Gonna have to disagree with you here, especially on the “I don’t think there would be large difference in number of anumals, only in number of species” aspect of your post.
Having travelled extensively to places (Manyika tribe in Zimbabwe and Ti Massaco tribe in Brazil) with much lower population counts, where they are still actively hunting and trapping as they were hundreds of years ago - if not thousands, I have opposing annecdotal evidence to the contrary.
Now, while I appreciate that these places today are known for their extensive individual populations of separate species of animals I must implore that these places are known for the aforementioned because they are relatively undisturbed by modern living.
Europe specifically would have seen MUCH higher densities (likely by a magnitude of 250%) of various subspecies of wild deer, sheep, hogs and boars, horses etc. People specialized in finding these animals and likely would have been able to track prey animals and predators alike such as wolves and bears to within a few hours of recent activity with relative ease. Rural living depended on it.
If you take into account that the red deer is prevalent enough today with modern development, I think it’s safe to say its numbers were magnitudes higher when more habitat was available to them - even with prolific predators keeping numbers capped to a certain degree.
And let’s not forget that deer farming was still a thing and actually, a very controversial method of obtaining deer products. Reading articles about this suggests that deer populations were actively managed and encouraged because to the common landlord, this was far more profitable than run-of-the-mill fruit and veg production in the short term; however the practise was looked down upon by those that could afford to look at the big picture economically and where the longevity of the health of the land was concerned.