I would encourage you to read the current literature. The evidence is more compelling than you might expect. Actually Meltzer’s 2020 paper (linked in my previous comment) touches on this exact concern under the subheading, “Getting past the Impasse”. There was another good paper that broke down how the end of the Pleistocene was radically different from previous glacial transitions and how the effects of that likely played into the extinctions. Combined with the revelation that most mammoth “kill sites” were actually more likely to be scavenged… it’s not at all an unlikely picture.