The fact is that these large, cold weather adapted mammals had survived numerous warm interglacial periods before, without going extinct. The difference then in those earlier interglacial periods was that either man did not yet exist, or later, man existed but did not have the very efficient tools and methods he would eventually develop about 15,000 years ago, or so. Without these, the mammoths and other large animals would still be around, because parts of North America, Northern Europe, and North Asia (Siberia) still resemble the climate and ecology which these animals inhabited during the ice ages; the only difference during the interglacial periods is that these ecological zones move northward. Mankind's better hunting skills were not going to make these animals extinct (necessarily) during the ice age, but during the difficult transition from ice age to interglacial period, the greater hunting skills of humans would have been enough to push several species into extinction when they were most vulnerable. Had ancient man had some kind of modern land management system (get your hunting license to hunt mammoth, etc.), then these animals would still be around. Instead, man overhunted them until they were gone (because they were easier to hunt or had more meat), and then switched to other species which were not as easy to hunt (and did not have as much meat) but which were better suited to survive man's predations.
And no one should buy this guy's "disease" argument. Where is the evidence that any large animal species was ever made extinct by disease? Sure, introduced diseases wiped out large numbers of American Indians, but they did not go extinct. They eventually developed resistence to these diseases, and survived. Animals are the same, and they do not live in large cities or other overpopulated clusters (unless domesticated by humans), so they are much less vulnerable to epidemics. This "theory" is nonsense, designed to deny the obvious effect that humans had on their environment; these politically correct types want you to believe that "environmental damage" is something that only rich white guys with modern technology are capable of doing. Rubbish. Mankind has been doing "environmental damage" ever since he discovered fire. It's an inevitable byproduct of human improvement; on the whole it was worth the price. No doubt some of the greenie-weenies disagree with me.
There are a couple of cases in which gold being discovered in a region led within a short space of time to the construction of a number of new churches in the region, as well as to new whorehouses. People like yourself who didn't know how to look past first-level correlations naturally assumed that religion was the root cause of immorality.
Other than that, the first time in the history of the world that humans ever had the combination of firepower and mobility to even think about exterminating animal species off entire continents was the advent of Chengis Khan's army. To try to picture American Indian ancestors doing anything like that is a sick joke.