Also, some of the things you mention as not mattering actually do matter very much to private school parents. We believe that they make a genuine difference to education in the broader sense - growth from learning, not just exam results.
small classes of academically able kids: you say that large classes don't do grammar school kids any harm. They might not harm their exam results (these are bright kids) but it absolutely changes the dynamic and the ability to discuss, develop debating and critical thinking skills, and make the class work at the best speed for all the students, leaving more time for things outside the curriculum.
Facilities don't need to be gorgeous and souped up, but if you believe that lots of exercise is critical for children's physical and mental development, and that games teach important emotional and teamwork skills, then you will consider sufficient sports facilities to be pretty important.
Likewise, a full theatre isn't necessary, but children learn confidence from frequent performances (musical and drama) so you do need sufficient dedicated space that these won't impinge on the rest of the school (decent sized hall with a stage is enough). And when they get older, they learn independence and resilience as well as technical skills by taking over the whole of the production of plays including lights and sound. And for personal growth, you want there need to be small enough numbers for all the kids to participate fully, and regularly, not just once in their school career.
And of course, sufficient staff to make all these things possible. That's the main cost.
Sure - all of these things are available outside school. And many middle class parents sign their kids up to classes to do all these things (VAT free - since they are considered education).
But having that within the school day - not just once, but as a regular, intrinsic and normal part of their school day - is part of the educational provision of private schools. If you strip it away, then you are genuinely removing education.
Yes, you're probably still left with a calmer, more disciplined environment, and that's important for some kids.
But it's a huge loss. And I believe that that extra education does genuinely make kids better than they would be otherwise. And I believe that does benefit the country at large: exactly as it benefits the country at large for a state school student to study the piano, or join a netball club, or go to Stagecoach, or do a drama summer camp.
Why remove education from kids? It doesn't benefit anyone else for those kids to not have these opportunities for their own personal growth, paid for by their parents.