Except that in cases where the Ice Age was long over, when humans arrived, the megafauna was still wiped out in a matter of a few hundred years at most. Madagascar and New Zealand were both first reached by humans well into historical times, with no significant change in the climate around that point - but bang, the megafauna were gone.
Even in the more well-known two cases (Australia and the Americas) they didn't happen during a common period of Ice Age. Humans reached Australia during the harshest part of the Ice Age, while humans reached the Americas when the Ice Age was ending. The two aren't really comparable in terms of potential climate stress, or much else.
Every time, so far as we know, the common factor is humans arrive, megafauna gone. (Or most of it- a few species did survive). There's no need to invoke climate stress as an additional factor, except insofar as climate changes allowed humans to reach regions (eg northern Europe, maybe North America) which they previously couldn't access.
Incidentally, even if new evidence comes to light which indicates that people were around longer before the megafauna died, it still doesn't mean that humans were incapable of hunting out big animals. It bugs me when people argue that humans couldn't hunt a large species to extinction. They damn well could; there's no species of land animal which humans are incapable of killing if they put their minds to it. The animals which survived appear to be those which bred too quickly to be wiped out, not ones where the individual animals were too hard to kill.
Also, in passing, there was no need for humans to have actually killed the large predators directly. Maybe they did, maybe they didn't, but it wasn't necessary. If humans wiped out the large animals which were prey for the predators, then the predators would go extinct anyway.
Oh I quite understand.
For the record I lean towards humans being primarily responsible for megafauna extinction, just not exclusively so.
I just get annoyed when people argue X v Y instead of looking at X in conjunction with Y and possibly Z etc.