Take away fire and starches/grains and you take away civilization. Grazing is for extracting calories from marginal land. Prime land isn't given to animals to graze on - it's given to plants. Once we knew how to harvest corms and tubers and cooked them we became more starvation proof. (Hunting is more hit or miss).

Starch granule are found in the teeth of many hominid fossils. We produce amylase in our saliva which breaks down cooked starches when it hits our tongues. Other omnivores do not have this adaptation.

You cannot have a high fat diet hunting (deer and small game simply do not have high amounts of fat). We moved on to agriculture because it provided plenty of calories and didn't run away from us when we tried to kill it.

While we may have eaten fat opportunistically - it's starches and fire that got us to where we are.