The human brain uses a lot of energy. It requires a lot of feeding.
Without any positive selection pressure it seems unlikely that such an "expensive" organ would evolve purely as a side product of other factors in the way I think you are suggesting.
That might be how it started out. But some sort of selection for bigger and bigger brains does seem to be required to explain the result we have ended up with.
Sexual selection seems a viable candidate. Big brains are sexy.