I really don't believe public schools are a significant factor for reducing child abuse. Family members, friends, and neighbors stop most child abuse.
"Also, there is plenty of child abuse even with public schools kids. Teachers are trained to recognize such, catch some now and then, but still much goes unnoticed."
There are also situations where teachers themselves are abusive, at least verbally/emotionally. I know of a serious case of this locally (the teacher was booted into administration). Then there are also the bad situations that can and do happen among peers on campus... I tend to wonder, simply based on our past school experiences, whether more often parents have to act to help their children in bad situations on campus than teachers need to act to help children in bad family situations.