It is very likely that the inability to adapt to rapid and radical climate swings had more to with the extinctions than human predation.
Yet there was more than one glaciation, or ice age, over the eons. Dozens just as long, just as cold, and just as quickly it became warm again. After 10,000 years or so, another ice age.
During all these many ice ages, the mammoth, the mastodon, the giant ground sloths, the Irish elk, the wooly rhinoceroses and other creatures stayed with us.
So why did they die off 10,000 years ago and not before?
Humans.
In Australia, the giant wombat, giant monitor lizard, a kangaroo over 9 feet tall, and the marsupial lion all died off about 40,000 years ago. Right around the time the first aboriginals showed up.
Even few in number, what happens if your way of hunting is to simply burn down the entire forest and eat what ever you want, already cooked? Prehistoric man had fire. There are fossil finds all over the West of mountains of bones under a cliff face—where prehistoric Indians drove the game animals. Giant bison, Saiga antelope, American camels, all a pile of bones here and there; all extinct.
Only in Africa were the megafauna not effected. And Africa, incidentally, is where humans evolved and animals in Africa have had millions of years to develop a healthy wariness of humans. The other part of the world where humans lived the longest—South and Southeast Asia, still have their megafauna, too. Indian Elephants, Indian Rhinos, tigers, panda... all learned to avoid scrawny, chatty but dangerous humans since the days of Homo Erectus.