Except the NCAA and its members insist players are students, not employees.
That's why I said similar, not identical.
What we're really saying here is that sometimes - like when it comes to paying athletes and providing them with protections such as workers comp - they're students. And other times - like when enforcing "non-compete agreements" and other restrictions - it's OK to treat them like employees.
Whichever is most beneficial to the institution.
No, that's not what we're saying here at all.
First, I think I've adequately shown that the need for some reasonable restriction is not "arbitrary" as you called it.
Second, the definition of "most beneficial" to the institution would be to give the institution 100% control over which teams (if any) a player can transfer. So, no, this is not a situation of "whichever is most beneficial to the institution."
The compromise I suggested seems completely reasonable and accommodates the competing interests of both the school and the player.