The first university was different than the tens of thousands that came after it.
The University of Bologna, founded in Italy in 1088, was run by students.
Eager to learn, the students came to Bologna from all over Europe. They hired the professors, and the professors weren't allowed to skip classes. They were required to start and end class on time. If fewer than five students came to a class, the professor was fined under the assumption that the class wasn't very good.
Professors weren't putting up with this for very long, and by the middle of the twelfth century the University of Paris, the world's second university, was organized around faculty. The system has stayed that way every since -- for good reason.
But it's still an interesting thought experiment: what would be different if your athletes paid your salary?