You're just outsourcing the authority you accept to some other group of adults and institutions. They don't have some special moral high ground that makes them better than a child's own parents and non-school social circle. If you care about your kids and have self confidence in your own character, why wouldn't you hold yourself up over strangers?
If the child sees nothing but their parents they have no parallax view to assess and learn for themselves. At a public school they will experience many adult teachers & see which they respect for which reasons. Not respecting your teachers but having to get along anyway is also a thing. It just seems horrific to restrict a child to a single relationship that the parents have such control over. Monstrous.