I was privately educated (assisted place as I did very well at 11+, this was over 40 years ago).
The private schools were, unfortunately, Catholic. They were also not very good academically, but I was allowed to quietly work at my own pace in the library for some lessons and I worked myself through Oxbridge entrance. I was very aware however of a lot of lifestyle differences (my family was effectively homeless when I started at private school for example).
The parents were often small businessmen, who were buying a notion of a genteel education. Also, their children were being educated into the idea that they were better than the kids at the comprehensive down the road (this was actually said). I remembered the kids from my state primary and I thought to myself that this was highly unlikely to have been the case.
When I went to Oxford, I found that I liked the bright kids who had gone to state schools, generally, better than the ones from public schools, who tended to be more up themselves (big generalisation of course). In particular, the arrogance often born of attending an elite institution really didn't appeal. Therefore, I did not want to send my own kids to private schools, in part because I worried about ending up with entitled kids who wanted to compete with their peers in terms of lavish lifestyle and with values different to mine, whom I might therefore come to dislike very much. (again it is of course more complicated than this, and I am less socialist than I used to be).
I think there are issues with bullying in both private and state schools, but I felt particularly unhappy about how we would navigate those seas with a private school.
In the end my children have gone to a good local village school, where they are doing well, and my son is likely to go to the nearby grammar. We are a bit out of catchment but he excelled at the 11+ and will probably get in. I don't like the ethos of a superior education based in large part on parental ability to produce cash, but I do like the ethos of a grammar school education, based, as far as possible, on natural ability. I think this will very much suit my son, and that is the most important thing at the end of the day.
I also realise that not everyone is so lucky as to be in the catchment of a good primary, or to have a shot at a grammar, or to have a child who could get on with the 11+. I wish we had more resources given to education.
And of course, private is a large financial commitment.