Harvard earns enough off trust interest they'd never have to charge another student tuition and room/board if they didn't want to.

They could just take all the smartest kids like this regardless of where they come from.

They do, technically. The admissions process is need-blind, meaning they decide whether to accept a student before looking at any financials, then provide enough financial aid for them to attend. Something like a quarter of the students don't pay anything.