Not expecting parents to pay won’t change things much for most students at most schools, because in most cases schools just don’t have the money to give away. The EFC is just a number with very little meaning. At best more students would qualify for a handful of federal and state grants, but then again it’s likely the qualifications for that would tighten because there is only so much money.
The only people that would benefit from making it a student only contribution (call it an “ESC”?) would be those at a meet’s full need school. But I suspect if that were the case, they wouldn’t guarantee to meet need for very long!