There is a great deal of argument over how effective early Man was at hunting, particularly larger animals.
One end of the spectrum is your notion, that animals provided a relatively small part of his diet.
The other end has the first men to arrive in America exterminating most of its megafauna within a few centuries of arrival. The Clovis extinction idea. To be fair, the current is running at present against this theory. Similar extinctions occurred more recently in New Zealand, Australia, Madagascar, etc.
My POV is that in every environment on earth that we know of historically Man was The apex predator. Pygmies regularly hunted and killed elephants and Eskimos hunted and killed polar bears without firearms. I suspect our prehistoric ancestors were fully as capable.
Oh they were.
Australia used to treed, from coast to coast, but the original inhabitants burned them all down in hunting the animals.