The athletes are already trying to unionize and, in part, claiming to be NCAA employees in some cases. I'm not an attorney but it seems the NCAA and Conferences make the rules, not the individual schools, so who would be the athlete's employer?
Basically, they're saying the NCAA controls the sports and employs the workers and they have a point that the lawsuits are all aimed at the NCAA, who then has changed the rules at the schools.
In the pro sports that's how it works, I think. The athlete negotiates their salary with the individual team but the player's union negotiates the collective bargaining agreement (pensions, salary caps, etc) with the league.
I expect college to follow that model when they are declared employees.