I have a solution for this. All these schools could become genuine charities and teach pupils for free, as many of their founders intended. (Eton and Chethams for certain sure.) They could be supported by generous donations from Tories, on which tax relief would be available. Winner, winner chicken dinner.
Except that if anyone could attend for free, these schools would no longer be bastions of privilege, would they? And that's what this is all about.
Perhaps they could also have reverse selection - taking the pupils most in need, as a normal charity would do.
It always 'amuses' me, that the 'best' state school near me, is a church school, and has pages and pages of entrance requirements. The criteria ensure that people go to Church and help out, mostly to get their kids into the school, while also being onerous enough to indirectly select less poor kids.
I sometimes wonder if that's what Jesus would have wanted?