Focusing on North America here:
One of the key food sources of the large mammals- the protein-rich forbs -- did not fully recover to their former abundance. This likely proved fatal for species like woolly rhino, mammoth, and horse in Asia and North America.
Protein rich northern meadowland forbs remained important constituents of the landscape in many large areas of the continent, including some areas that had supported these megafauna throughout the last three glacial advances and retreats. Several species of megafuana dependent on them survive still (moose, several species of deer, antelope) - and several species not apparently dependent on them vanished suddenly (giant sloth, giant beaver, teratorns, big cats and bears and wolves, etc).
The review concludes there is only firm evidence for about 8 to 14 megafauna species still existing when Aboriginal people arrived. About 50 species, for example, are absent from the fossil record of the past 130,000 years.
Circumstantial evidence of that kind is worth pondering, but in North America a short gap in a fossil record that spans a glaciation (landscape altering event on a huge scale) is not solid.
Of the remaining 36 species of Australian megafauna that vanished for some (obviously unusual) reason(s) in that 130k window, how many are absent from the very recent fossil deposits by simple accident of geography, ecological pattern, or human search?
Besides: the disappearance of 14 species - including entire genera, not just varieties - of continental megafauna in a few thousand years is a notable event in itself. The fact that the event likely includes a share of 50 others simply highlights.
YD et impact event anyone?
Influential by presumption, but too short, too small, too localized, to universal in its influence, and not severe enough to explain a continental extinction wave among megafauna only. The glaciation itself they had survived - a far more significant climate cooling event than the YD.
Another possible event recently supported with some indicative evidence would be a significant meteor impact on the remaining ice sheet. Again, this does not explain the species distribution or scale of the extinction wave, but it would have added severity to other pressures - boosted the odds.
The pivotal observation, however, is this one: every well-known and well-investigated human colonization of a formerly human-free land area of this planet has been followed by a wave of rapid and human caused extinctions among the large resident warmblooded fauna. The pattern is without exception, and has solid if complex mechanistic explanation supported by empirical observation. The presumption, the null hypothesis, the conventional basis of organization, would have to be that the pattern extends to those human colonizations we still know little about.
If the discussion is about the range or scope of some group of extinctions, or the fate of some particular animal that seems to have been different, or the exact mechanistic sequence of events (such as the advent and effects of widespread landscape burning by humans, the effects of rats, etc), or the roles of other circumstances such as climate fluctuations, then that is one thing. If the argument is that humans colonized some region of the planet without wiping out any significant portion of the large warmblooded fauna, then we need to see some serious evidence and sound reasoning. How did that happen?
Note that the Australian attribution to climate change for most does deal with mechanism - the climate change was continent wide, the fire regime supposedly predates human firing (this depends on the date of human colonization, of course - the window is not large, for the natural fire regime advent), and the aborigines supposedly had no adequate hunting gear (a dubious claim that casts a bit of doubt on the whole article - Kalahari bushmen hunt and kill African elephants, which are large and intelligent and well-experienced with humans, using gear no more "adequate").