I don't see how scientists could argue it one way or the other definitively.  The answer may be "both" rather than one or the other.

I don't see how the food thing alone could be very persuasive. There are many species of primates with food preferences very much like ours and our ancestors' in the wild, and not all of them suddenly went all brainy.

There is a strong tendency for a certain style of worker to get a single idea that amounts to a rush of blood to the brain, and then to propound it as the next best thing to received Truth. Such ideas usually are simplistic at best by the very fact of their one-dimensional nature. Whatever led to your superb brain and mine, I am sure of one thing, and that its that it was no one thing, but involved multiple special circumstances, including genetics, ecology, and evolutionary opportunism in general.

I think our early ancestors did not eat much/many grains because the now common grains were much smaller and difficult to gather/process efficiently. Grain consumption became significant only after the hunter/gatherer stage.  The "stationary' lifestyle probably was due to the "domestication"  and agriculture of berries/fruits and and root crops.

The domestication of grains (which largely required selective breeding techniques learned from other crops/animals) certainly improved the stationary lifestyle.

One much contested theory of human evolution (google aquatic ape) is that humans were apes that specialiized in living along the shores of rivers, lakes or seas. Perhaps this started with foraging along these shores with their rich sources of easily gathered animal life (mussels, crustaceans and washed up fishes.) Perhaps this positive feedback loop explains the sudden and fast evolution of human "intelligence" which in these circumstances required "the opposable thumb", tool making and fire, to find and process these treasures.

I wander (sic).