Again, you are making a broad claim outside the consideration of mitigating factors, like habitat.
89 primate species out of around 230–270 eat meat. Most of the species that do eat meat get those calories from insects, spiders, grubs, eggs, and fish. They also live mostly in tropical environments where larger animals (except other predators) are not plentiful. Effectively, there’s nothing larger to hunt.
Once you get outside tropical environments, you still have primate species who hunt smaller game but you also have species that are known to hunt larger prey.
Bonobos are known to take down duikers, which are about half their body weight. Baboons will hunt small antelope. In these more temperate environments you still have a larger variety and availability of small game and insects as well as a larger variety of nutritious plant life.
When you get closer to savanna type habitats like early hominids lived in, you find a shift in hunting patterns again. The plant life in these areas contain many more tough grass species that are not digestible to omnivores. They also have less selection and availability of bugs, grubs, eggs, and fish.
The end result is that food availability pushes hunting behaviors.
Prey list of common chimpanzees and Olive Baboons: forest duikers, impala, bushback, bushpigs, other primates (Galago spp.)
In the end, all predatory and omnivorous species are going to predominantly prey upon whatever prey species are most available with the least amount of effort. So evolutionary pressure that leads a species to select larger prey is more likely to do with what’s available and whether or not the body’s digestive system can make use of it than it does providing more “fat calories for a larger brain".
Not when bodies produce their own fat. Not when there are many species with physically larger brains (though not proportionately) than humans who are able to feed those big brains on plankton or grass, bark, roots, and fruits.
Of course, size isn’t the only consideration of brain evaluation. Number of neutrons and firing capacity are also important. On that note, while it’s true that humans have a higher encephalization quotient than extant animals, it is not out of line for a primate, even compared to the grub eaters and fruitarians.
It’s intellectually dangerous and misleading to look at evolutionary pressures in two dimensional this or that/this leads to that sorts of terms. We live in a three dimensional world where survivability and species success are challenged by overlapping evolutionary pressures that play off of and against each other in unique and wondrous ways.
