Not capitalist. Just local control - as in neighborhood, township, home, parish, whatever. Schools don’t have to be corporately funded for profit institutions. Profit isn’t God. Used to be a lot of schools were non-profit, started by religious philanthropists.
At this point, the smaller the better.
Religious schools fall into the capitalist system. They are privately owned and run. Capitalism is very encouraging of charity and religion. Church owned and run schools would be a major sector of the market if we did away with socialized schooling.
I’m assuming, of course that you aren’t talking about a government church.