You do realize that the school isn't the one who gets to make the choice, right? We aren't talking about intramurals.
In order to play games, schools need other schools. They can't play themselves. To do so, they join leagues with other schools. In high school sports they join conferences who are in turn members of the overall leaque that administers rules for their members statewide. This organization sets rules on how to play the game, how many practices are allowed, who can coach, who is responsible for paying officials, etc. A large chunk of these rules involve player eligibility. This includes academic requirements, transfer rules, age limits, weight classes, banned substances, medical eligibility and gender/sex issues.
I guess a school could allow a player on the team that was ineligible, but that means they forfeit every game that that player plays in.
Privatizing such a system isn't going to change that.