user12931: It may be hard to find a comparative monastic tradition in the west although there are some monastic in Christianity currently most are really assimilated and intertwine with the Church which bears its existence in charity; so it's not practical to compare these with Buddhist monastic tradition. However, if we go back in history there were monastics in Christianity completely detached from the church, for example, the desert fathers ( Christian ascetics in Egypt and Syria ) were completely detached from the church, but  following the Apostel's command that 'a person unwilling to work should not eat' they lived by the labour of their hand.
So, the question, in short, is why do Buddhist monks rely on alms for sustenance? 
I don't mean to sound sarcastic, but for many observers, it's perplexing to see a young and strong monk asking for sustenance from old and ragged and gray women.

user-1: As i see it,
as an ascetic one has no interest in anything else than realizing the Dhamma. If say one in training decides to practice extreme asceticism to starvation for reasons other than thinking that it is great thing to do but maybe circumstances make one choose between a livelihood that severely impairs the practice or starvation one could make a case for it being at least seemingly a better option relatively speaking, i dont think it is optimal but it is always an alternative. One has some 1-2 months or even more if one brings stored food to get as much work done as possible.
I mean that it is in a way a burden to not sever the connection to laylife completely. More to illustrate if say one person became got some attainments in time when and where it is prosecuted and given the option to go his own way or to be hindered by circumstances. He might choose to just go do his thing and not teach anybody anything.

user12749: The monks are suppose to develop concentration as much as possible and concentration demands a life with no preoccupations, if they had to work, it could undermine their proficiency in concentration. Also, they have to keep practicing satipatthana uninterrupted, if they had to work, it could provoke lapses of mindfulness, which would undermine their gain of insight. Not to mention, lay people supporting the sangha is a opportunity to practice generosity and make merit.

user10100: Gharāvāsa (layman) = ghara (house) + vasa (living) = the person who live in the house with 5 kāmaguṇa, the cause of disturbance, such as color, sound, smell, taste, and touch. And if you even read M.N. Mūlapaṇṇāsaka mahādukkhakkhandhasutta, 5 kāmaguṇa also included the lay-style-job  for sustenance, too.
Samaṇa (monk) = sama (peace) + ṇa (person) = the person who live in peace, freedom from 5 kāmaguṇa.

Reasons for laying down the course of training for monks; Purposes of monastic legislation (every vinaya rules included these 10 purposes):
“Because of this, monks, I will lay down a training rule for the monks
  for the following ten reasons: 
A. For the advantage of saṅgha: 

for the comfort of the excellence of the unanimous Order.
for the comfort of the Order.

B. For each monk's personal advantage:

for the control of shameless persons.
for the living in comfort of well-behaved monks.

C. To be the protector:

for the restraint of the cankers in the present; for the prevention of temporal decay and troubles.
for warding off the cankers in the hereafter; for protection against spiritual decay and troubles.

D. For the faith of the community:

for the confidence of those who have not yet gained confidence.
for the increase of the confidence of the confident.

E. For the advantage of buddha's teaching:

for the lastingness of the true doctrine.
for the support of the discipline.


So, buddha made the rule such as money-getting-disallowing rule for the monks. 

user471: From this guide for laypersons interacting with monks:

The Buddha suggested that the basic source of food for bhikkhus
  was that received on the morning alms round (pi.n.dapaata). This daily
  dependence on alms food reminds both the bhikkhus and the lay devotees
  of their interdependence and prevents the bhikkhu from becoming too
  isolated from the lay community. He 'meets' them every day and eats
  the food that they share with him. Several important rules are
  concerned with this as well as a major section of the Sekhiya Training
  rules.
An alms round is not considered begging, for the bhikkhu does not
  solicit anything but is ready mindfully to receive any alms that lay
  people may wish to give. Although alms food may sometimes be meager,
  the bhikkhu is always expected to be grateful for whatever he is
  given. It is surprising how particular we can be about what food
  we like to eat; and what complications that can cause. This is
  reflected in the way rules concerning 'edibles' are arranged, which
  may seem very complex especially when the bhikkhu's life is supposed
  to be so simple. It should be borne in mind that the rules often deal
  with extraordinary circumstances and try to prevent them from becoming
  the norm.

However, there are the exceptions called the Great Standards which may allow monks in non-Buddhist countries to bend the minor rules by cooking or farming:

Already during His lifetime, the Buddha made special allowances for
  different regions (or desa) outside the 'Middle Country' of North
  India — where He lived and taught. These dealt with both the workings
  of the Community — for example, a smaller quorum for ordination is
  allowed in distant parts where there are fewer monks — and practical
  measures, such as special dispensation for footwear and bathing. (See
  EV,II,p.173) So there is a precedent for adapting to conditions, but
  this does not mean the abolishing of any rules [see End Note 6].
The Lord Buddha also left us a set of principles that can still be
  used as a standard to judge new circumstances. These are known as
  'The Great Standards.' Properly used they should protect against a
  wholesale dilution of the Rule.
This is how the Great Standards are formulated:
"Bhikkhus, whatever I have not objected to, saying, 'This is not
  allowable,' if it fits in with what is not allowable, if it goes
  against what is allowable, that is not allowable for you.
"Whatever I have not objected to, saying, 'This is not allowable,' if
  it fits in with what is allowable, if it goes against what is not
  allowable, that is allowable for you.
"And whatever I have not permitted, saying, 'This is allowable,' if it
  fits in with what is not allowable, if it goes against what is
  allowable, that is not allowable for you.
"And whatever I have not permitted, saying, 'This is allowable,' if it
  fits in with what is allowable, if it goes against what is not
  allowable, that is allowable for you." (BMC p.27; see also EV, II,
  p170)
Treated with care, these Great Standards should enable bhikkhus to
  live according to the Vinaya Rule in, for example, isolated
  communities in non-Buddhist countries with non-tropical climates. They
  form a touchstone for modern conditions and substances.

The Accayika Sutta talks about the most urgent duties of a monk:

"There are these three urgent duties of a farming householder. Which
  three?
"There is the case where a farming householder quickly gets his field
  well-plowed & well-harrowed. Having quickly gotten his field
  well-plowed & well-harrowed, he quickly plants the seed. Having
  quickly planted the seed, he quickly lets in the water & then lets it
  out.
"These are the three urgent duties of a farming householder. Now, that
  farming householder does not have the power or might [to say:] 'May my
  crops spring up today, may the grains appear tomorrow, and may they
  ripen the next day.' But when the time has come, the farming
  householder's crops spring up, the grains appear, and they ripen.
"In the same way, there are these three urgent duties of a monk. Which
  three? The undertaking of heightened virtue, the undertaking of
  heightened mind, the undertaking of heightened discernment. These are
  the three urgent duties of a monk. Now, that monk does not have the
  power or might [to say:] 'May my mind be released from fermentations
  through lack of clinging/sustenance today or tomorrow or the next
  day.' But when the time has come, his mind is released from
  fermentations through lack of clinging/sustenance.
"Thus, monks, you should train yourselves: 'Strong will be our desire
  for the undertaking of heightened virtue. Strong will be our desire
  for the undertaking of heightened mind. Strong will be our desire for
  the undertaking of heightened discernment.' That's how you should
  train yourselves."


user111: I recommend you to read the Kasi Bharadvaja Sutta. Kasi Bharadvaja was a farmer who asked this exact question from the Buddha. Monks dedicate their lives to spiritual development. They do not have time to make money or do agriculture. They do not eat for sport or beautification. They only eat enough to maintain life so that they can work towards Nibbana. Yes, farming does break a precept.
"Should any bhikkhu dig soil or have it dug, it is [an offence of Confession.]" 

user-1: The single most important reason why Buddhist monks do not traditionally support themselves is to put them forever dependent on the laity.  The Buddha did not want monks isolated from householders.  By requiring them to make daily alms rounds, it put them in constant contact allowing them to preach and be exemplary models of the holy life.  If monks were to live in isolation and make their living as farmers, it would be all too easy for there to be an unbridgeable schism between monastic and lay practitioners. 

12931: <p>It may be hard to find a comparative monastic tradition in the west although there are some monastic in Christianity currently most are really assimilated and intertwine with the Church which bears its existence in charity; so it's not practical to compare these with Buddhist monastic tradition. However, if we go back in history there were monastics in Christianity completely detached from the church, for example, the desert fathers ( Christian ascetics in Egypt and Syria ) were completely detached from the church, but  following the Apostel's command that 'a person unwilling to work should not eat' they lived by the labour of their hand.</p>

<p>So, the question, in short, is why do Buddhist monks rely on alms for sustenance? </p>

<p>I don't mean to sound sarcastic, but for many observers, it's perplexing to see a young and strong monk asking for sustenance from old and ragged and gray women.</p>
 -1: <p>As i see it,
as an ascetic one has no interest in anything else than realizing the Dhamma. If say one in training decides to practice extreme asceticism to starvation for reasons other than thinking that it is great thing to do but maybe circumstances make one choose between a livelihood that severely impairs the practice or starvation one could make a case for it being at least seemingly a better option relatively speaking, i dont think it is optimal but it is always an alternative. One has some 1-2 months or even more if one brings stored food to get as much work done as possible.</p>

<p>I mean that it is in a way a burden to not sever the connection to laylife completely. More to illustrate if say one person became got some attainments in time when and where it is prosecuted and given the option to go his own way or to be hindered by circumstances. He might choose to just go do his thing and not teach anybody anything.</p>
 12749: <p>The monks are suppose to develop concentration as much as possible and concentration demands a life with no preoccupations, if they had to work, it could undermine their proficiency in concentration. Also, they have to keep practicing satipatthana uninterrupted, if they had to work, it could provoke lapses of mindfulness, which would undermine their gain of insight. Not to mention, lay people supporting the sangha is a opportunity to practice generosity and make merit.</p>
 10100: <p><strong>Gharāvāsa (layman)</strong> = ghara (house) + vasa (living) = the person who live in the house with <strong><a href="http://www.84000.org/tipitaka/read/v.php?B=12&amp;A=2784&amp;Z=3013&amp;eng=buddhavacana" rel="nofollow noreferrer">5 kāmaguṇa</a></strong>, the cause of disturbance, such as color, sound, smell, taste, and touch. And if you even read M.N. Mūlapaṇṇāsaka <a href="http://www.84000.org/tipitaka/read/v.php?B=12&amp;A=2784&amp;Z=3013&amp;eng=buddhavacana" rel="nofollow noreferrer">mahādukkhakkhandhasutta</a>, <strong>5 kāmaguṇa also included the lay-style-job  for sustenance, too.</strong></p>

<p><strong>Samaṇa (monk)</strong> = sama (peace) + ṇa (person) = the person who live in peace, freedom from 5 kāmaguṇa.</p>

<blockquote>
  <p><strong><a href="http://www.84000.org/tipitaka/dic/d_item.php?i=327" rel="nofollow noreferrer">Reasons</a> for laying down the course of training for monks; Purposes of monastic legislation (every vinaya rules included these 10 purposes):</strong></p>
  
  <p>“Because of this, monks, I will lay down a training rule for the monks
  for the following ten reasons: </p>
  
  <p><strong>A. For the advantage of saṅgha:</strong> </p>
  
  <ol>
  <li>for the comfort of the excellence of the unanimous Order.</li>
  <li>for the comfort of the Order.</li>
  </ol>
  
  <p><strong>B. For each monk's personal advantage:</strong></p>
  
  <ol start="3">
  <li>for the control of shameless persons.</li>
  <li>for the living in comfort of well-behaved monks.</li>
  </ol>
  
  <p><strong>C. To be the protector:</strong></p>
  
  <ol start="5">
  <li>for the restraint of the cankers in the present; for the prevention of temporal decay and troubles.</li>
  <li>for warding off the cankers in the hereafter; for protection against spiritual decay and troubles.</li>
  </ol>
  
  <p><strong>D. For the faith of the community:</strong></p>
  
  <ol start="7">
  <li><strong><em>for the confidence of those who have not yet gained confidence.</em></strong></li>
  <li><strong><em>for the increase of the confidence of the confident.</em></strong></li>
  </ol>
  
  <p><strong>E. For the advantage of buddha's teaching:</strong></p>
  
  <ol start="9">
  <li>for the lastingness of the true doctrine.</li>
  <li>for the support of the discipline.</li>
  </ol>
</blockquote>

<p>So, buddha made the rule such as <a href="http://84000.org/tipitaka/pitaka1/v.php?B=02&amp;A=2567&amp;Z=2684&amp;eng=suttacentral" rel="nofollow noreferrer">money-getting-disallowing rule</a> for the monks. </p>
 471: <p>From <a href="https://www.accesstoinsight.org/lib/authors/ariyesako/layguide.html#alms" rel="nofollow noreferrer">this guide</a> for laypersons interacting with monks:</p>

<blockquote>
  <p>The Buddha suggested that the basic source of food for bhikkhus
  was that received on the morning alms round (pi.n.dapaata). <strong>This daily
  dependence on alms food reminds both the bhikkhus and the lay devotees
  of their interdependence and prevents the bhikkhu from becoming too
  isolated from the lay community.</strong> He 'meets' them every day and eats
  the food that they share with him. Several important rules are
  concerned with this as well as a major section of the Sekhiya Training
  rules.</p>
  
  <p>An alms round is not considered begging, for the bhikkhu does not
  solicit anything but is ready mindfully to receive any alms that lay
  people may wish to give. Although alms food may sometimes be meager,
  the bhikkhu is always expected to be grateful for whatever he is
  given. It is surprising how particular we can be about what food
  we like to eat; and what complications that can cause. This is
  reflected in the way rules concerning 'edibles' are arranged, which
  may seem very complex especially when the bhikkhu's life is supposed
  to be so simple. It should be borne in mind that the rules often deal
  with extraordinary circumstances and try to prevent them from becoming
  the norm.</p>
</blockquote>

<p>However, there are the exceptions called <a href="https://www.accesstoinsight.org/lib/authors/ariyesako/layguide.html#modern" rel="nofollow noreferrer">the Great Standards</a> which may allow monks in non-Buddhist countries to bend the minor rules by cooking or farming:</p>

<blockquote>
  <p>Already during His lifetime, the Buddha made special allowances for
  different regions (or desa) outside the 'Middle Country' of North
  India — where He lived and taught. These dealt with both the workings
  of the Community — for example, a smaller quorum for ordination is
  allowed in distant parts where there are fewer monks — and practical
  measures, such as special dispensation for footwear and bathing. (See
  EV,II,p.173) So there is a precedent for adapting to conditions, but
  this does not mean the abolishing of any rules [see <a href="https://www.accesstoinsight.org/lib/authors/ariyesako/layguide.html#fn-6" rel="nofollow noreferrer">End Note 6]</a>.</p>
  
  <p>The Lord Buddha also left us a set of principles that can still be
  used as a standard to judge new circumstances. These are known as
  'The Great Standards.' Properly used they should protect against a
  wholesale dilution of the Rule.</p>
  
  <p>This is how the Great Standards are formulated:</p>
  
  <p>"Bhikkhus, whatever I have not objected to, saying, 'This is not
  allowable,' if it fits in with what is not allowable, if it goes
  against what is allowable, that is not allowable for you.</p>
  
  <p>"Whatever I have not objected to, saying, 'This is not allowable,' if
  it fits in with what is allowable, if it goes against what is not
  allowable, that is allowable for you.</p>
  
  <p>"And whatever I have not permitted, saying, 'This is allowable,' if it
  fits in with what is not allowable, if it goes against what is
  allowable, that is not allowable for you.</p>
  
  <p>"And whatever I have not permitted, saying, 'This is allowable,' if it
  fits in with what is allowable, if it goes against what is not
  allowable, that is allowable for you." (BMC p.27; see also EV, II,
  p170)</p>
  
  <p>Treated with care, these Great Standards should enable bhikkhus to
  live according to the Vinaya Rule in, for example, isolated
  communities in non-Buddhist countries with non-tropical climates. They
  form a touchstone for modern conditions and substances.</p>
</blockquote>

<p>The <a href="https://www.accesstoinsight.org/tipitaka/an/an03/an03.091.than.html" rel="nofollow noreferrer">Accayika Sutta</a> talks about the most urgent duties of a monk:</p>

<blockquote>
  <p>"There are these three urgent duties of a farming householder. Which
  three?</p>
  
  <p>"There is the case where a farming householder quickly gets his field
  well-plowed &amp; well-harrowed. Having quickly gotten his field
  well-plowed &amp; well-harrowed, he quickly plants the seed. Having
  quickly planted the seed, he quickly lets in the water &amp; then lets it
  out.</p>
  
  <p>"These are the three urgent duties of a farming householder. Now, that
  farming householder does not have the power or might [to say:] 'May my
  crops spring up today, may the grains appear tomorrow, and may they
  ripen the next day.' But when the time has come, the farming
  householder's crops spring up, the grains appear, and they ripen.</p>
  
  <p>"In the same way, there are these three urgent duties of a monk. Which
  three? The undertaking of heightened virtue, the undertaking of
  heightened mind, the undertaking of heightened discernment. These are
  the three urgent duties of a monk. Now, that monk does not have the
  power or might [to say:] 'May my mind be released from fermentations
  through lack of clinging/sustenance today or tomorrow or the next
  day.' But when the time has come, his mind is released from
  fermentations through lack of clinging/sustenance.</p>
  
  <p>"Thus, monks, you should train yourselves: 'Strong will be our desire
  for the undertaking of heightened virtue. Strong will be our desire
  for the undertaking of heightened mind. Strong will be our desire for
  the undertaking of heightened discernment.' That's how you should
  train yourselves."</p>
</blockquote>
 111: <p>I recommend you to read the <a href="https://www.accesstoinsight.org/tipitaka/kn/snp/snp.1.04.piya.html" rel="noreferrer">Kasi Bharadvaja Sutta</a>. Kasi Bharadvaja was a farmer who asked this exact question from the Buddha. Monks dedicate their lives to spiritual development. They do not have time to make money or do agriculture. They do not eat for sport or beautification. They only eat enough to maintain life so that they can work towards Nibbana. Yes, farming does break a precept.</p>

<p><em>"Should any bhikkhu dig soil or have it dug, it is [an offence of Confession.]"</em> </p>
 -1: <p>The single most important reason why Buddhist monks do not traditionally support themselves is to put them forever dependent on the laity.  The Buddha did not want monks isolated from householders.  By requiring them to make daily alms rounds, it put them in constant contact allowing them to preach and be exemplary models of the holy life.  If monks were to live in isolation and make their living as farmers, it would be all too easy for there to be an unbridgeable schism between monastic and lay practitioners. </p>
