I can only speak for one university, the University of Maryland.
UM keeps funding and admissions decisions completely separate.
That is, first you have to be admitted. Then they decide if, and how much, funding they will offer you. But I think it's based on merit, not need.
In other words, I think they don't really care how much money you have or don't have, they only care how qualified you are, for BOTH decisions.
That's how I understand U of MD to work; I always assumed that most U.S. universities work that way, though I could be completely wrong.
John