Q:

Why would you prefer employer to pay all non-taxable benefits and you pay for the benefits that are taxable if your employer pays them?

Tax Planning for You and Your Family 2019 by KPMG. p 153.

10.1.1 Non-taxable employment benefits
Certain employment benefits are not taxable, even though many of them are
  deductible expenses to the employer. The government is thus offering an
  incentive to employers to provide these benefits, since the after-tax returns
  are greater than straight salary. Non-taxable benefits include:
• Contributions to a registered pension plan (the pension is taxable
  when received—see 3.5.1)
  • Contributions to a group sickness or accident insurance plan
  • Contributions to a “private health services plan”, such as those
  covering drugs, medical expenses and hospital charges not covered
  by public health insurance, and dental fees (except for Quebec
  purposes—see 17.10.2)   

p 154 [Let me know if you want me list whole page of bullets]

• All or a portion of the cost of free or subsidized school services for your
  children (for example, if the services are provided in a remote area)
  • Contributions to a supplementary unemployment benefit plan

p 155. I omit six benefits at page top.

Note that where your employer deducts its contributions to a sickness
  or accident insurance plan, a disability insurance plan or an income
  maintenance insurance plan, any benefits you receive from the insurance
  will be taxable (but reduced by your own premium payments to the plan).
Ask your employer to make use of the nontaxable
  benefits outlined above as much as
  possible. If you have an employment benefit
  package where you and the employer share the
  costs, try to have the cost-sharing reallocated so your employer pays for
  all the non-taxable benefits and you pay for the benefits that are taxable if
  your employer pays them. In many cases, without changing the cost to your
  employer, you can reduce your tax burden this way.

Don't you want reverse? Isn't non-taxable supposed to be swapped with *taxable** in that sentence?
Isn't employer supposed to pay for all TAXABLE benefits so that you aren't taxed? Wouldn't you prefer paying all NON-TAXABLE benefits because you don't pay tax? 

A:

Isn't employer supposed to pay for all TAXABLE benefits so that you aren't taxed? Wouldn't you prefer paying all NON-TAXABLE benefits because you don't pay tax?

No, because you are misunderstanding exactly what taxable benefits means: possibly because in the above sentence you're only emphasising the taxable bit and perhaps overlooking the benefits bit).
A taxable benefit is something an employer provides on which you need to pay tax as though they had given you the equivalent value of that benefit in cash as part of your salary. On the other hand, a non-taxable benefit is one where you don't need to pay tax (in effect, the equivalent value is deducted from your salary before it assessed for taxes).
Note: I'm neither a Canadian, nor an accountant. The examples below may not be 100% accurate and are for illustrative purposes only. They only consider federal taxes, based on the figures from Canadian income tax rates for individuals - current and previous years. Specifically, that for 2019 the tax-rate on taxable-income of up to just shy of $48k is 15%. To keep figures simple, I've also taken the basic personal amount (the amount you can earn before tax) to be $12k (instead of $12,069 for 2019).
Suppose you earn $48,000 per year. Your employer offers you an extra $500 per month as either (a) a non-taxable benefit (e.g. contribution to an appropriate pension plan); (b) a taxable benefit (perhaps a company car1); or (c) extra cash in your pay-packet.
Case A: Non-Taxable Benefit
Gross income:            48,000
Basic personal amount:   12,000-
                        -------
Taxable income:          36,000
Tax payable:              5,400-    (15% of 36,000)
                        -------
Post-tax income:         30,600
Non-taxable income:       6,000
                        -------
Total income:           $36,600

Case B: Taxable Benefit
Gross income:            48,000
Taxable benefit:          6,000
Basic personal amount:   12,000-
                        -------
Taxable income:          42,000
Tax payable:              6,300-    (15% of 42,000)
                        -------
Post-tax income:         35,700
                        -------
Total income:           $35,700

Case C: Salary Increase
Gross income:            54,000     (48,000 + 12x500)
Basic personal amount:   12,000-
                        -------
Taxable income:          42,000
Tax payable:              6,300     (15% of 42,000)
                        -------
Post-tax income:         35,700
                        -------
Total income:           $35,700

As you can see, having the employer pay for a non-taxable benefit leaves you $900 per year better off, compared to having them pay for a taxable benefit (or simply giving you the cash). This figure is (of course?) 15% of the $6,000 value of the benefit/raise.
In simplistic terms2, if you were to pay for non-taxable benefits, you would be doing so out of already-taxed-money, whereas your employer does so out of pre-tax money (thus reducing the amount you do pay tax on).

1 A company car might not be the perfect example (in the UK the tax situation on them can be, I believe, complicated). If necessary, substitute anything that does not fall into the non-taxable category.
2 Depending on how similar (or not) the Canadian system is to the UK's, it's possible/probable that you can claim back the tax already paid on non-taxable expenditure.

