You cannot give athletes a stipend. The current NCAA President tried to pass a $2,000 a month per diem and it was turned down by the majority of NCAA institutions. Why?? Because while Athletic Depts make "millions", they also spend "millions" on athletes.
Everyone thinks athletic departments roll in revenue, but the truth is very few turn a profit and football is used to subsidize the other sport programs. The vast majority of athletic departments borrow money from the academic side.
Also, go ahead and pay football players since some of their programs turn a profit --- then get ready to see women collegiate athletes in court over Title IX sex discrimination/benefits lawsuits along with possibly non-revenue producing male sports.
Lets say the women win (and also the non-revenue producing sports for men) --- how will universities cover the cost to pay these athletes?
Hike tuition on the normal student on the academic side so a women's tennis player (3rd string punter, lacrosse player, etc) that no one cares about can have a scholarship, pell grant money, housing, food, book costs covered, etc with $2,000 a month while the academic student majoring in the same thing is forced to pay even more.
Paying college athletes is Pandora's box and it's going to be a massive train wreck on collegiate sports.
EDIT
Also, would paying student athletes really stop players from taking extra benefits? Look at the accusations recently --- luxury cars, $1,000 suits, 10K for autographs. Would 1K or 2K a month curtail that?
Everyone pushing pay for play harps on the student athletes making money for the universities (Peyton, Manziel, Tebow), but what about the student athletes costing the university money? Anyone dying to have current third teamer jerseys? Anyone care about them other than potential "depth" until they are recruited over and seemingly disappear?