Q:

How did the talmud's description of stoning work in practice?

I am trying to understand the description of stoning given in Sanhedrin 45a (starting here).  To summarize:  They take the condemned to a place that is twice a man's height, the first witness pushes his hips to cause him to fall on his face,  and if that didn't kill him they turn him over and the second witness drops a (prepared, heavy) stone on his chest.  If that doesn't kill him either then all assembled stone him (method not specified).  This description raises questions for me.
Was the push expected to often be sufficient to kill the condemned?  The g'mara a littler farther down says that they aimed for a compassionate death, so if the fall usually wouldn't kill him that seems at odds with this goal.  But it seems implausible that a fall from a height of a mere 12 feet or so would usually be fatal, even onto a hard surface like stone, though a direct blow to the head can change things.  (Complication: we are not told whether, and if so how, he is restrained.)  
Perhaps the fall is not expected to kill him and the heavy stone is usually needed.  This involves both witnesses, which complies with "the hand of the witnesses shall be upon him" (D'varim 17:7).  On the other hand, they could have had both of them push him if that were the only consideration.  (The g'mara discusses the stone being so heavy that it took both of them to move it, so the witnesses already worked together somewhat.)
So my question is: when in this process did the condemned usually die, and if it was sometime after the fall, why do we have at least a two-step process instead of something that would kill him faster and with less pain, such as just placing him on the lower surface and then dropping the boulder on him?  (Not raising the platform; the g'mara is concerned about excessive height causing disfigurement.)

A:

The statement in the gemara is that the height was usually sufficient to kill the person immediately. Note that he was standing when he was pushed so that his head was at three times the height of a person (about 17 - 20 feet). This would usually be enough to break his neck. Note that even a fall of ten tefachim could be enough to kill him. Dropping the stone on him from the height would actually be less certain as he may move and it may hit him wrong.

If a fall of ten handbreadths is sufficient to kill a person, why must
  the platform from which the condemned man is pushed be twice the
  height of an ordinary person?

Halachically, this means that if an animal dies from a fall into a pit of this depth, then the person who dug the pit is legally responsible, because it can be a fatal drop. Having the person stand on a platform of a full height of twice the height of a man (6 amos) is used to ensure that it would be fatal in most cases.
This then means.

Therefore, even though the one being executed is likely to die from a
  fall from a lesser height, a platform is built that is twice the
  height of an ordinary person in order to ensure a quick and relatively
  painless death.

The average height of a person is about 3 amos. Thus the head of the person being executed is about 9 amos above the ground. The gemara also indicates that most times the dropping of the second stone was not required. However, if it was required, the fall was enough to ensure that the stone would be immediately fatal.
Wikipedia estimate a tefach as 3.16–3.76 in so that ten tefachim is on the order of 40 inches. It says that an amah is 6 tefachim so that 6 amos is 36 tefachim. The criminal's head at about 9 amos would be 54 tefachim. The wikipedia estimate is 18.96–22.56 in which is close to two feet, making the persons head almost 18 feet above the ground. 
And then Art Scroll Sanhedrin 45a4 notes 33 and 39 explains that this is the procedure required based on the pasuk in the Torah.

33 i.e. by being pushed off a precipice. 
 Up to this point the Baraisa has spoken only of the penalty
  to be administered to those who touched the mountain during the
  revelation at Sinai. The Baraisa now teaches that the same procedures
  are to be followed in every case the Torah mandates the penalty of
  sekilah

These notes apply to the following gemara.

The mishna teaches that one of the witnesses who had testified against
  the condemned party pushes him off the platform. Concerning this
  halakha the Sages taught in a baraita: From where is it derived that
  the punishment of stoning can be fulfilled by pushing the condemned
  party from a high place, so that he dies from his fall? The verse
  states with regard to those who crossed the boundaries that were set
  up around Mount Sinai and touched the mountain: “Take heed to
  yourselves, that you not go up into the mountain, or touch the border
  of it; whoever touches the mountain shall be put to death; no hand
  shall touch him, but he shall be stoned or shall be thrown down”
  (Exodus 19:12–13). And from where is it derived that this punishment
  can be fulfilled with actual stoning? The verse states: “He shall be
  stoned.”
And from where is it derived that this punishment is sometimes
  fulfilled both by stoning and by pushing, i.e., if the transgressor
  did not die from his fall, he is then stoned? The verse states: “He
  shall be stoned or shall be thrown down.” And from where is it derived
  that if the condemned man died from the pushing, the obligation to
  stone him has been fulfilled, and there is no further need to actually
  stone him? The verse states: “Or shall be thrown down,” with the term
  “or” indicating that only one of the two options must be fulfilled.
  And from where is it derived that this is the halakha not only at
  Mount Sinai, but even with regard to future generations?

