"If there was a mechanism for kids from different backgrounds to access those kinds of schools more easily, it might help with the class issue. I don't think it would make a difference with the sex question though."
There is. It's the bursary system. Eton is, in fact, one of the most generous for bursaries, paying for a couple of years at a prep school prior to admittance for some scholarships. Of course, there are the elite, rich and famous at these schools, but there are also some children who are from poor families, who are in need of the kind of education these schools offer.
It has meant that these children of the upper classes have heard the other side of the story: how Margaret Thatcher wrecked the unions from the grandson of a union man, how she affected mining communities from the family of a miner. It's an opportunity for them to learn, too.
These schools will never go away. The big public schools are well endowed, they have land. There will always be schools for the very rich. Do we really want to make them more exclusive and elite?
What you will likely do is to tear down girls education. Girls' schools are not nearly so well endowed. Some of the public schools are becoming co-ed: Charterhouse (Jeremy Hunt was head boy) is the latest, but the all-girls schools are not nearly so well off.
What would even things up far more is the same kind of investment into girls' education. There is no Eton/Winchester/Harrow equivalent for girls, with the same kind of money behind it.