From what I've read, such species as the wooly mammoth, the mastodon, and so forth were on their way to extinction before humans arrived in the NA. Climate change, from Ice Age to post-Ice Age, was one big reason cited for the decline. Humans helped, but were not the sole cause.

By the way, the same case can be made for the buffalo (bison bison). Not climate change, but brucellosis, with the "hunters" helping them along. And yes, I'm aware that the brucellosis was most likely brought in by cattle.