I went to a British Public School ( about as upmarket and private as you can get)
We still had , as Senior Prefects , the right to "fags". ie a young boy to run errands. In the winter in fact they were sent to lie in beds , to warm up the beds for prefects.
Caning was still allowed by teachers , up to 6 strokes even.
Hell my sister went to , at the time ,THE most expensive bording school in the world , and she said it was a hot bed (pun intended) of lesbianism.(OK Nuns were involved)
So private school gains a lot of advantages but little protection from child molestors