According to that same paleontologist, the Pleistocene was the backdrop against which a 'deadly duet of evolution and extinction' first played itself out between humans and elephants. That humans were responsible, at least in part, for the demise of the large mammals of the Ice Age ought not to surprise us. There are several modern cases in which overhunting by man has nearly exterminated a species. You're an American. Think of the American bison.
Mass extinctions are complex phenomena.
As the ice retreated at the end of the Pleistocene, the plant communities upon which mammoths depended became increasingly fragmented, less diverse, less able to support a diverse assemblage of animals. Many scientists believe that mammoths may have been in decline during the last four millennia of the Pleistocene and that the change to a warmer climate, combined with the stress of predation by man, pushed them over the edge. You mentioned the Neanderthals earlier. Ah, for that I'll have to turn you over to our visiting physical anthropologist. Maybe I should start by saying that we know what we know about them primarily from studying the fossil record, though techniques used in molecular biology have recently begun to provide us with some new information.