You're sticking to wooly mammoths. The Americas had quite a few proboscideans outside of 'em. In fact South and North America combined had more proboscidean species than Africa+Eurasia, most of which lived in template and hot climates and had short or no hair. And guess what, Africa+Eurasia only has lost one proboscidean in the last 10000 years (excluding dwarf island species), while the Americas have lost all.Climate change can't explain megafauna extinctions, especially when you take into account that megafauna living in temperate and hot areas went extinct as well, after living for millions of years there, and that in the case of the Americas the extinctions are more and more younger as you move to the south. Mastodons and ground sloths were still around in Argentina by 4000 bC, for example. That can only be explained through an overkill hypothesis. Humans entered the Americas from the North, and the extinctions followed the same pattern.
I beg to differ on this one; there's another, relatively recent, theory that suggests that humans, or rather, their dogs, ticks and flees, introduced a number of diseases in the New World and Australia - diseases to which the native animals did not have any natural resistance.
And there's already plenty of cases of animal populations and even entire species that were devastated or driven to extinction largely thanks to one or a few diseases in modern times - there's several documented cases of lion and African wild dog populations that were devastated by rabies, and an infectious disease caused by chytrid fungus is thought to be the main cause of the sudden extinction of a few dozen amphibean species around the world.
There's no reason why something similar could not have happened in prehistoric times - especially considering the fact that the American and Australian megafauna species evolved in (relative) isolation, and therefore did not come into contact with a good number of Old World diseases until the humans and their dogs and parasites came along.
And if you add the introduction of new, deadly diseases to animal populations that are already stressed by climate change and (over)hunting, then it's really not suprising if those animal populations go extinct fairly quickly.
Especially if they're populations of large, long-lived animals with slow reproduction rates, like the mammoths, mastodons and giant sloths.