The owners of a school or any business can place whatever rules they like in their property.I'm against public schools, but that's separate matter.
actually, having public schools is very beneficial for maintaining your dominance over the plebs. Cause now you can tell the children they also had the chance to go to school, just they weren’t smart enough to make it, thus having them believe the ****ty situation they were born into is their fault. If you don’t have public schools, they might actually notice something is not quite right when they can’t pay the school fees.
Edit: or am I missing something and you use „public school“ like the British do, confusingly meaning „private school“?