In theory you could. But, that argument falls apart when you realize that no athlete worth his salt would want to be the “14th man” on a stacked team. If he’s actually the 1st man, then whoever is the 14th man won’t stick around, thus opening up a scholarship. There’s only so many minutes available.
But more importantly, if school A offers a million dollars and you have to pay your own way while schools B through Z offer a million plus the full ride, School A has put itself in an unnecessary recruiting disadvantage.