Did climate change kill off the wooly mammoth?

I've never understood the claim that "climate change" killed off all the megafauna in North America except the bison just as the Indians arrived from Siberia. 

The View from Wilshire Blvd.: tragic baby mammoth

For example, an immense number of skeletons of extinct animals have been dredged up from the La Brea Tar Pits on Wilshire Boulevard next to the Los Angeles County Museum of Art: mammoths, mastodons, giant sloths, saber-toothed cats, camels, horses, and dire wolves. They all went extinct about the time that Indians showed up, as the Ice Age ended, allowing the Indians to break through into the main part of North America. I guess you could use it as a euphemism for how the Siberians were locked out of the center of North America by glaciers until "climate change" occurred, but the way it's usually used makes no sense.

During the Ice Age, the climate in Los Angeles was about like it is today in the Monterey Peninsula, 300 miles to the north. So, if the wooly mammoths started to feel uncomfortably warm in Los Angeles, why didn't they just walk to Monterey? They're elephants, right? Same with the camels. I could conceivably imagine giant sloths not being mobile enough to head north, but horses?

The most plausible explanation of the mass extinctions in North America about 11,000 years ago was given by Jared Diamond a couple of decades ago in The Third Chimpanzee:

THE UNITED STATES DEVOTES TWO NATIONAL HOLIDAYS, COLUMBUS Day and Thanksgiving Day, to celebrating dramatic moments in the European "discovery" of the New World. No holidays commemorate the much earlier discovery by Indians. Yet archaeological excavations suggest that, in drama, that earlier discovery dwarfs the adventures of Christopher Columbus and of the Plymouth Pilgrims. Within perhaps as little as a thousand years of finding a way through an Arctic ice sheet to cross the present Canada-U.S. border, Indians had swept down to the tip of Patagonia and populated two productive and unexplored continents. The Indians' march southward was the greatest range expansion in the history of Homo sapiens. Nothing remotely like it can ever happen again on our planet. 

       The sweep southward was marked by another drama. When Indian hunters arrived, they found the Americas teeming with big mammals that are now extinct: elephantlike mammoths and mastodonts, ground sloths weighing up to three tons, armadillolike glypt-odonts weighing up to one ton, bear-sized beavers, and sabertooth cats, plus American lions, cheetahs, camels, horses, and many others. Had those beasts survived, today's tourists in Yellowstone National Park would be watching mammoths and lions along with the bears and bison. The question of what happened at that moment of hunters-meet-beasts is still highly controversial among archaeologists and paleontologists. According to the interpretation that seems most plausible to me, the outcome was a "blitzkrieg" in which the beasts were quickly exterminated — possibly within a mere ten years at any given site. ...

We are all too familiar with the blitzkriegs by which modern European hunters nearly exterminated bison, whales, seals, and many other large animals. Recent archaeological discoveries on many oceanic islands have shown that such blitzkriegs were an outcome whenever earlier hunters reached a land with animals naive to humans. Since the collision between humans and large naïve animals has always ended in an extermination spasm, how could it have been otherwise when Clovis hunters entered a naive New World?

THIS END, though, would hardly have been foreseen by the first hunters to arrive at Edmonton. It must have been a dramatic moment when, after entering the ice-free corridor from an overpopulated, overhunted Alaska, they emerged to see herds of tame mammoths, camels, and other beasts. In front of them stretched the Great Plains to the horizon. As they began to explore, they must soon have realized (unlike Christopher Columbus and the Plymouth Pilgrims) that there were no people at all in front of them, and that they had truly arrived first in a fertile land. Those Edmonton Pilgrims, too, had cause to celebrate a Thanksgiving Day. 

But now we are told that Jared Diamond is some kind of horrible racist, so we can't listen to him anymore.

Or I could imagine that the Indians brought diseases with them, or another species like dogs or rats, maybe with some kind of plague fleas. Or maybe they set giant fires.

The last mammoths survived until about 1650 BC on Wrangel Island, in the Arctic Ocean, northwest of the Bering Strait:

Instead, Dalen and the rest of the team believes some drastic change must have occurred on Wrangel Island to kill off the mammoths, and there are two likely culprits: humans and climate. Archaeological evidence suggests that humans reached Wrangel Island at roughly the same time the last mammoths vanished, but there's no evidence yet to indicate that they ever hunted the mammoths. The more likely answer is climate change, which as a side effect might well have made it easier for humans to reach the island to serve as witnesses to the mammoths' final days.

Uh, sure. Who can't imagine the Siberians standing around saying, like an NPR announcer, "Oh, the tragedy of these  mammoths, but thankfully they survived just long enough for us to see them so we can treasure their memories."

Okay, some sort of climate change could have wiped out this population of 500 to 1000 mammoths that were stuck on an island the size of Delaware because they had nowhere else to go. But, this amazing coincidence that whenever humans show up anywhere, the mammoths die out at about the same time seems a bit unlikely. It's kind of like blaming the near extinction of the bison in the late 19th Century on some change in the weather rather than on Buffalo Bill and his colleagues.

But Native Americans are into, like, you know, ecology, so they couldn't have just hunted down the mammoths, right?

If the human newcomers in North America and Wrangel Island had been white men, would there be much controversy over whether or not they were responsible for the extinctions?