You can't have it both ways. If you are against the players getting paid due to the amatuerism argument. Then that should apply to the school too. Admission to games should be free. Players can't be used or required to participate in any sort of marketing or promotions. Jerseys can't sold using a players name on them. The Coach can't get paid for coaching the team only reimbursed for travel expense, you would have to employ them as a teacher in academic curriculum for them to get paid. The refs have to be volunteers etc. just like how they do with elementary school sports.
Do all that and then i get your point. Otherwise it makes no sense for the players to be the only ones considered amatuer under that logic.
If you can't do that, then the players should be compensated like everyone else is.