Most of the financial aid given out by these schools isn’t federal aid. They even give full rides to international students who aren’t eligible for federal at all, and middle-class kids whose families have too much income to be eligible. This aid is funded by the endowment which is donations from rich people.

I think when people say Princeton receives a lot of federal money, they are talking about research grants, and possibly the effective subsidy that comes from the tax deductible status of donations from rich people. Each of these adds up to a lot more money than the Pell grants.

If you take away the rich people, the school won’t be able to afford to operate this way.