I thought that was the consensus.

Harry had once read a famous book called Chimpanzee Politics. The book had described how an adult chimpanzee named Luit had confronted the aging alpha, Yeroen, with the help of a young, recently matured chimpanzee named Nikkie. Nikkie had not intervened directly in the fights between Luit and Yeroen, but had prevented Yeroen's other supporters in the tribe from coming to his aid, distracting them whenever a confrontation developed between Luit and Yeroen. And in time Luit had won, and become the new alpha, with Nikkie as the second most powerful...

...though it hadn't taken very long after that for Nikkie to form an alliance with the defeated Yeroen, overthrow Luit, and become the new new alpha.

It really made you appreciate what millions of years of hominids trying to outwit each other - an evolutionary arms race without limit - had led to in the way of increased mental capacity.

'Cause, y'know, a human would have totally seen that one coming.

HPMoR, chapter 24

I'm having a tough time seeing how the gene-level selection works in your account, though. Genes don't inherently care about memes [...]

Memes have benefits to genes. Copying avoids the costs of individual learning. It allows you to obtain lots of good ideas quickly, by copying them from successful individuals. Good ideas are important. They can be mastery of fire, hunting techniques, love songs or negotiating strategies. Thus we have meme-spreading adaptations: our incessant babbling, our ultrasociality.

and larger cranial capacity comes with major disadvantages that require overwhelming genetic selective advantages to compensate.

These costs apply equally to any theory of human cranial expansion. Not much besides the enormous benefits afforded by cultural inheritance can reasonably expect to pay for them.

Certainly our big brains are the result of a sexually-selected arms race - but what was sexy was a GSOH, being able to sing love songs, etc. IOW, the products of cultural inheritance.