Most institutions don't "make money on sports", in the sense that their athletic departments operate at a loss, which if it can't be covered by alumni donations or by taxes for public institutions must be extracted from students. (College administrators might not see a difference between the latter situations and "profit"...) Like many other aspects of the modern university, a huge portion of athletic expenses are incurred by employing many times the number of administrators required in decades past. The NCAA functions on the athletic side in much the same way that the bankruptcy-impervious student loan functions on the academic side: ratcheting up revenue to support the inexorable growth of administrative staff. Colleges and universities aren't doing a better job than they did years ago, so by charging more they defraud both students and society.

Whether students are allowed to profit or not, I certainly support your proposal to ban colleges and universities from taking commercial interest in student sports.