The OP's question focuses on Garner's remark:
Because the former is the correct reading, some believe that notwithstanding should be sent to the end of the phrase in which it appears.
The (extremely dubious) purist argument is that
An outpouring of editorial opinion on either side of this issue notwithstanding, there are no easy answers.
The outpouring of editorial opinion did not withstand the absence of easy answers.
Notwithstanding an outpouring of editorial opinion on either side of this issue, there are no easy answers.
The absence of easy answers did not withstand the outpouring of editorial opinion.
The "correct reading," Garner observes, is that the attempts by editorial opinion writers to persuade everyone that the answer is clear-cut fail to resolve the underlying complexity of the issue; that is, the editorial arguments don't withstand the force of the opposing thesis that there are no easy answers. As a matter of structural logic, the "notwithstanding-at-the-end" construction makes this point, while the "notwithstanding-at-the-beginning" construction does not.
Part of the reason this analysis seems a bit awkward is that Garner's example involves nested oppositions: there is the opposition between the argument (by editorial opinion writers) that the problem at issue has easy answers and the counterargument (made by the writer) that there are no easy answers; but there is also the opposition between editorial opinion writers who think one answer is obviously correct and editorial opinion writers who think that the opposite answer is obviously correct.
The example's nested oppositions muddy the waters somewhat—so Garner presents a less complex example of what the logical purists are talking about. As he notes, the purists' reading of "notwithstanding" as meaning literally "not withstanding" leads them to conclude that the following sentence is logically correct:
The family's objection to the marriage notwithstanding, [the marriage took place].
because the objection did not withstand the couple's desire to marry, as proved by the fact that the objections did not prevent the marriage.
In contrast, the purists would say, this sentence is illogical:
Notwithstanding the family's objection to the marriage, [the marriage took place].
because it states that the family's objection was "not withstood" at the same time that it clearly shows that the couple did withstand the objection by getting married anyway.
One might argue this it is a very odd purist indeed who insists on the critical importance of proper placement of "notwithstanding" to indicate which of two opposed sides successfully withstood the other, and yet seems to care very little that use of "not withstanding" is not a very logical choice of terms in a situation where "not preventing" would characterize the situation more accurately. After all, the family's objection to the marriage may still stand—and remain as firm as ever—whether the couple married or not.
A true logical stickler might, therefore, insist on reserving "notwithstanding" for instances in which something literally failed to withstand something else. For example:
The barbarian horde eventually sacked the town, the town's stout defenses notwithstanding.
But all of this malarkey takes place in a world of prescriptive logicality that bears little connection to the the real world of English as she is spoke. As Garner points out, writers have been using "notwithstanding" in the "illogical" way for more than 600 years, strongly suggesting that English writers and speakers have never acknowledged the logical rule that the purists wish to promulgate.