We get rid of athletic scholarships altogether. The schools can begin offering student employment positions as the players of different sports, which they are free to make as lucrative as they publicly want to be. The contract can be seasonal, meaning that people can't just pick and choose different schools every game. However, players do have to be students at the college, so transfers can be somewhat controlled by the schools.

Everyone has 5 years of eligibility, which they are free to divide up however they want.