The objective of the very best players is not to merely pay for college, but maximize their chance to join professional teams. Those odds are maximized by access to world class facilities, equipment, trainers, coaches, medical care, and more. Winning helps too.
If some schools have less money because sponsorships are going to specific players instead of the school/league, they will be able to provide fewer top-tier resources to players. Consequently, those schools will be less attractive in the areas of essential importance to the very best players.
If the very best athletes gravitate toward non-player-sponsorship schools, it will create a self-reinforcing cycle to get other top athletes, because they want the best teams to maximize their chances of winning.