This is something I agonise over. I do think the OP is partially correct that private schools do entrench privilege and division in society at some level. But I also privately educate my child. Which makes me a massive hypocrite.
I could provide a long of reasons to justify my decision to privately educate my child which I feel are legitimate but I won’t because it’s boring special pleading.
The point @LolaSmiles and others are making is correct, though, and one which means it will be almost impossible for the OP and others to win this argument.
And this is that all parents who are ambitious for their children or care about their education at some level are involved in entrenching this privilege: this isn’t solely the preserve of parents that go private.
It’s the parents who spend an additional £150k to move into catchment for a good comprehensive. It’s the parents who stick with the comprehensive and tutor their kids or who socially engineer in order to make sure their kids have the “right” friendship groups. Or the ones who muscle in on the governors.
None of this is wrong, per se. But it all serves to entrench privilege and widen inequality. And parents who do all this and pat themselves on the back because they stop short of private schools are not being honest with themselves.
I grew up in a town which boasts a comprehensive which has some of the best A level results in the country. People are paying vastly over the odds for houses in catchment. But their consciences are all clean because they don’t (technically) go private (though they do all tutor)
I went to a private school in the same city. Loads of my contemporaries who went to this comprehensive still tease me about having been to private school. And it’s utter hypocrisy because they were all the recipients of vast amounts of privilege but in such a way that their parents social democratic credentials survived.
I don’t think any decision is “wrong” as long as it’s above board and done with the children’s needs in mind. But I do think people who have gamed the system to remain just the right side of the socialist purity line but lavishing a lot of social capital on their kids ought to maintain a dignified silence around other parents attempts to give their children the best chances they can within the private sector.