I depends how far back in time you refer to. Availability of large animals meat is likely to have become commonplace only after our ancestors had developed hunting strategies, techniques and weapons, which is relatively recent on an evolutionary scale, just like or even posterior to the control of fire.
We are way better adapted to eat large animals and even "cultivated"ones like sheep, beef etc that have been around for long, than eating the cultivated Agri products you find in shops today. It is pretty clear by now what fructose does to our health. And how dangerous it can be to go vegan.
It is very easy to catch fish and mussels/oysters, crabs... for much of that you do not need anything else than your hands...
Didn't the Native Americans eat a majority of buffalo and were completely healthy? I'd have to agree with sabertooth about quality over variety.
They were decimated by infectious diseases brought by the Europeans and these Indians would probably haven’t been harmed by these if they were on properly balanced 100% raw diet.
We all know how dangerous foreign viruses and bacterias can be for native populations. It has nothing to do with them eating a not optimal diet. They are just so totally not adapted to stuff from another continent. Like we also would easily get issues in Africa's Jungle without having modern medicine. I have heard about instincto's dying from malaria down there....
That said, I have read the white man also gave the Indian's alcohol and other stuff that must have made them more sensible to disease too.
What I have read about the American Indians, they were just stunningly healthy and strong, their wounds healed so fast the white man was shocked.
From what I know today, this fact does not surprise me at all.