the nuts and bolts of that will doubtlessly need to be worked out (assuming the NLRB decision isn't reversed in November when the Trump nominee takes over).
But, in the abstract, an employer can absolutely stop an employee from going elsewhere in certain circumstances. If an employee is under contract, for instance, they cannot just up and join a competing company (i.e. Tom Brady can't decide he wants to be close to his parents and play for the 49'ers in the middle of a contract). Another way is a non-compete clause that some employees sign (though the scope, strength, and enforceability of those vary by jurisdiction.
Should this NLRB decision stand, I imagine the private schools will just have to come up with some patch. Perhaps contract language, where in exchange for the scholarship and related expenses the athlete must play at the school? That does put into play some fun scenarios. Is it a 4 year deal with a school option for a 5th year? If they are contracts, then private schools could potentially trade players :noidea:Popcorn and soda though I'm sure for public perception reasons the contract would stipulate it can't be assigned elsewhere.