Responses to comments

I agree that private schools suck up the best teachers. It seems unfair if one imagines that very good teachers can improve any pupil, even if all other factors remain the same.
But if we take that assumption, then simply having different quality teachers is unfair, no matter where they are. So we can't really solve the problem entirely, although forcing teachers to work where we choose in order to keep schools' teaching abilities level will reduce the inequality.
On a practical note, that's just what people have tried to do to junior doctors, and the doctors are outraged. The good doctors don't see why they should suffer bad jobs just to satisfy unpleasant patients; good doctors have the strange belief that as better doctors they might do better in the better jobs, and make more difference there than struggling in almost hopeless jobs.
And the government has recanted and scrapped its disastrous system.
I think that teaching is similar. It's not simply that a good teacher will get the same amount more out of any pupil, but that a good teacher can get a bigger improvement from a good pupil, whereas some bad pupils will not respond to any teacher. I'd also suggest that there's a limit to what good pupils can learn from bad teachers.
All of which means that good teachers have more effect with good pupils, so it makes sense to put them there. Of course, private schools don't take all the good pupils, even if they have extensive bursary schemes; but were the private schools not providing some selection, the good teachers would be mostly wasted, especially if they are required to spend more of their time with bad pupils. Private schools are clearly not an optimal solution, but they're better than not having private schools. The optimal solution would be to bring back grammar schools, which although they do miss some people, would enable almost all the best pupils from any background better to fulfill their ability.
But the currently fashionable opinion is that grammar schools are a bad thing; so much so that no political party supports them any more.