This is the benefit I mentioned above. Since it is federal aid, it can only go to students the school can qualify and the easiest way to do that is using the FAFSA since only qualified students (citizens, resident aliens) can file it, it requires a SSN and verification with an FSA ID. It is just an easy way for the schools to verify the student is eligible.
Schools also can loosen the requirements for work study if they have extra funds available, so even students with high EFCs get approved.