School doesn't make people more intelligent. It makes them more educated, which can mean a variety of things depending on the cultural context, such as an understanding of social mores, skills such as reading and writing or arithmetic, and/or ideological indoctrination.
Most other things in the human body seems to be use it or lose it. Sure, innate talent matters a lot, but so does training. You’d never be a track star even with the best genetics if you never left the couch.
that's the problem though, the mind isn't a muscle. lifting weights can make you better in a different but related activity, like a sport. but the research on brain games shows that they only make you better at the specific game they're drilling, not anything else.