But your hypothesis fails to explain why similar large animals continued to survive elsewhere in the world, i.e. Africa. Why did the elephants there and in Asia survive human predation? Regardless, I think this was some darn fine detective work on the part of these guys.
By the way, elephants are endangered in Africa. It's not the climate that's endangering them.
Actually, the fossil record shows that extinctions happened pretty much everywhere when people first showed up. The large, slow, docile animals were killed off, and only the ones that were either fast enough to elude us, strong enough to confront us (ever seen a pissed off elephant?), or bred fast enough to replentish their losses survived. The giant sloth and armadillo were easy pickings and had lots of meat, so they went first. Larger animals like the mastodon bred so slowly that their numbers simply dwindled under the constant human culling. The previous apex predators like the cave bears and sabre tooth tigers died off simply because we hunted out their food sources. There was nothing left for them to eat other than the faster animals, and the leaner cougars and wolves were simply better suited to that kind of hunting. There's absolutely nothing far-fetched about any of this, so I don't see why it's so controversial.
If you move into a foreign area and have to live off the land, are you going to work your butt off hunting down a fast antelope with a heavy spear, or are you going to kill the tasty sloth that can only run 3MPH?
When humans moved in, we became the new apex predator and we altered the ecological balance accordingly. This is simply the way the world works.