Here's a very well-written article from 2020, from the respected D. Meltzer, that breaks down the evidence really concisely and cites most of the other literature I would point to: https://www.pnas.org/doi/10.1073/pnas.2015032117
All of that said, I agree with and want to add to your main point. We did see commercial hunting result in exactly the conditions you described in early American history. Actually, famous commercial hunters like Boone recognized that we were quickly losing the prevalence of wildlife (although it is likely that the prevalence was only there because of the recent demise of the majority of the Indian - and I use that word for the same reason as the author I'll cite - population that had been hunting them, see Charles Mann's "1491").