Private schools are in a difficult position. They have no remit to interfere with what goes on in their students' private homes. My DC was at an independent secondary on a scholarship, DH works at one.
Whilst a majority of students and parents were lovely, there were clearly many parents who did not parent their children. But at the same time some of those parents are too involved in orchestrating their child's friendships - many which will have been formed as their parents got friendly with other parents at the joining prep-school. It always looked like they were full of angst and buying popularity for their teenagers by throwing exclusive parties for them, alcohol featured from year 9.
In my opinion there is much more that private schools could do, and some of these are personal to my DC's school but likely to be more universal
Bring in more extra-curricular activities which foster co-operation and not just competition
Stop giving sports team such hero status. My DD tells me of the times a boy has been sanctioned for bad behaviour in class, only for it to be downgraded when he is need to play an A team rugby match 
There are external organisations connected with sports that could be brought in to address boys and give them new role models and a better expected code of conduct.
Stop making feminism a dirty word. If girls want to set up a feminism club, don't make them change the name and tie yourself in knots to make it sound like it is a club to address the equality of everyone
I think some of this behaviour is more prevalent in private schools because the leaders of those schools had a similar education themselves.