My father went to one of the oldest English “public” (i.e. private) schools. en.wikipedia.org/wiki/Latymer_Upper_School He didn’t talk about the academics, which is surprising for an academic- he talked about the antisemitism he faced every single day from kids, teachers and staff. I’m sure it didn’t help that his parents were poor and he was there on scholarship.
I went to a private school in the U.S. for elementary school. I was bullied every day, not just by the kids, but by the only teacher I had from first through sixth grade, and he terrified me so much that my parents didn’t know until I was an adult and my mother ran into another kid I went to school with who talked about how sorry she felt for me.
My daughter goes to an American public school. She is bullied a lot too (we’re an eccentric family), but at least the teachers are mostly on her side, and if one isn’t, I have someone to complain to about it. I wouldn’t even think of risking her in a private school.