Certainly that's a good question, and the two themes interact. Most private schools have some sort of relgious core, typically CoE.
DS1 is at a highly selective private school, and you'd have to do a really crap job not to have them leave with good results. Being a Christian school, they avoid having many special needs to screw with their statistics.
That is an issue the unions don't really address.
Most private schools are selective, quite openly.
DS1's school friends are all above average academically, and since the school selects on behaviour as well, they're nice in a slightly spooky way.
Does this harm society ?
Certainly teaching unions would say so. They want smart kids in their scools to dilute the harmful effect of the Sky Sports generation. But I don't want my kids used to help undo defective parenting.
Also, in our case the superstition is not hard line. Although there's a full time chaplain it doesn't ban evolution or persecute any of the many kids whose parents aren't superstitious, or who have a different invisible friend to compete with Jesus.
Ask your4self. If a school says "Jesus is the good guy@, what does it say about the kids who deny his very existence ? Think they have a good time from the other kids or from the teachers themselves ?
The nearest school to us is CoE which, with a fanatic in charge. Even Ofsted who see their job as PR bunnies for schools had dark remarks about the bullying problem and how in spite of it being an area of essentially zero deprivation and near 100% use of English at home, the results were very mediocre.
I would go hungry to pay fees rather than let my kids go there. And yes, before you ask I have gone hungry back when we were poor.
The Chaplain at DS1's school is not a bad chap, and we've had vigorous arguments over religion. One left my hand bleeding where he threw a piece of metal at me. No I don't know why I caught it rather than duck
Private schools are a far smaller % of system than you might think, and way lower than state schools driven by people who think "Narnia" is a historical textbook.
My kids don't meet any working class kids through school at all. Even competitve sports seem to be solely between private schools. They do meet a pretty broad range of backgrounds, and given that the parents aren't failures, they will experience meeting admirable, but very different people.
Education is compulsory, and if you don't have the resources, then you have to take what the state gives you. Superstion based schools increase the distance between schools and homes causing both lower outcomes and pollution.