Perhaps the non-Etons that are flailing should consider going down the state route. I've to go now but I'd like to thank everyone who engaged with my question - it really was asked out of curiosity and from a not fully informed position.
That said, I do find the A level debacle to be a clear manifestation of unfairness in educational access, and I am really saddened at many many of the current and last governments' policies in relation to education and childhood inequality. I lived in the UK under Labour and benefitted greatly from sure start centres, one o'clock clubs and the whole infrastructure that seemed to be dedicated to creating a better start for all children, and less alienation for parents. I trust the research that concludes equal societies are better for absolutely everyone, and (in other countries) it's a policy area I've worked in.
To my mind, it seems that while charitable status may technically describe some activities of these schools, the reasons for them to maintain that status isn't really defensible in terms of the spirit of what a charity should be - and I don't think that it can be justified by the argument that removing it would then mean more students would then be in the state sector - so what? Then fund the state sector better, spend energy on lobbying for that.
I also don't think offering occasional use of their facilities justifies it - if their charitable status is predicated on their contribution to education the use of a swimming pool isn't within the remit of that. The idea that they turn out well educated citizens doesn't really wash when it's such a tiny minority of usually already wealthy people that then acquire positions of power due to their parents' ability to pay huge fees.
I don't care if people here send their children to private/public schools (the nomenclature is tricky), a lot of people would like to be able to ensure their children are privileged and have an excellent education, but I really don't find any of the arguments for charitable status compelling or reasonable. Thanks again