You are correct that currently there is nothing that should prevent any of that.
But, this is now professional sports by definition - the act of being paid for doing a job. These aren't regular students or normal working individuals - they are professional or semi-professional athletes.
Every organized athletic organization in the world where compensation is involved has a set of rules and limitations that are far different from normal schmucks like us. If you want to treat the kids like semi-pro athletes now, then why shouldn't there be the same restrictions in place as any other league? Things like salary cap, trade limitations, contracts, free agency, etc.
I realize none of that infrastructure is there today, but I don't necessarily see paid athletes as normal people, just like I don't see NBA, NFL, MLB, Soccer, etc players as normal people. They make the choice to be bound by the agreements and structure above them. If they don't like those limitations, they aren't forced to be bound by them. They can elect to work in the real world where freedom is the rule.