Roughly half the public money for "research" is actually pocketed by the university and spent on education/administration/etc (the university calls it "overhead"). This is a dirty little secret of science funding - a big chunk of it isn't funding science at all, but is merely a subsidy for big research colleges.

Also, even tuitions at private colleges are massively subsidized - the subsidies just follow the students (e.g., subsidized student loans, need-based aid, etc).