The vast majority of schools use SAI to figure out how to prioritize the limited need based aid they give, but don’t necessarily meet need. For example, students with unmet need may be eligible to compete for funds (need based aid) that students with SAIs above cost of attendance are not.
Over award rules are for students that have full cost of attendance covered through grants and scholarships (aka full rides). The out of pocket for the student is still zero whether the school is giving all of the financial aid or if they reduce the package by what the student’s pell eligible amount is to recoup that. Even if no pell is involved, an outside award (like a local community scholarship) would reduce the aid the school could give by that amount. Still, the student pays nothing, whether their SAI was 0 or 100,000. But, full rides are rare.