The actual cost for a kid to matriculate is really rather small. After all they're going to pay the professor the same to teach 20 kids as 21. Only a very few get full. DD has Presidential +$2500 from engineering so my cost will still be around $7-10K/yr. The only cost consideration comes from the amount they're NOT getting; if they're getting enough without these kids contributing all is well.
So yes, the guaranteed scholarship was the primary reason Bama was initially considered. (Rolling admissions helped too.) Without that, there are plenty of fine colleges that would have gladly taken my $25K.