picc, it's all incredibly difficult.
My parents had working class backgrounds (one's father was a bus driver, the other's a communist warehouseman whose CV was complicated by having been a political conscientious objector on WW1). They went to grammar schools under the 44 act and then a redbrick. So they are the winners of selection. The 11+ was abolished in Birmingham two years ahead of me and they sent me to a local comp, rather than the various local residual grammars. The middle-classes were so glad to see the back of the 11+ that my parents' decision was common, and I don't think I'd have done any better anywhere else: I too went to a good redbrick.
But the grammar schools were colonised by the middle-classes, with selected others allowed in on sufferance. The system was all about identifying talent and moving it into universities, which sounds terribly egalitarian, but delivered nothing to those that didn't get into university. And a marxist analysis would be that the effect of the 1950s education system was to remove working class children, who might have presented a challenge to the class system, and buy them off with middle-class jobs that neutered their ability to improve the interests of their class. You may laugh, but a variation of that argument is sometimes used to justify not encouraging bright children to apply to elite universities. The problem with meritocracies is that they quietly ignore the effect of those who don't have the means to achieve in the competition.
But the middle-classes keeping the aspiration in the comprehensives relies on a certain anti-grammar solidarity. Once a few of them defect (to use the language of the prisoners' dilemma) everyone has to follow. Now, the middle classes are less likely to be present in the local comp for reasons of social conscience (they might be at primary, which is less stratified) and more likely to either colonise selected schools which they bend to their will, or legacy free grammars, or independent schools.
And every proposed solution fails. One that had some popularity on the left was to tie places at elite universities to deprived schools, so that it was much, much easier to get in than from elite schools. Sounds good? OK, I'll enrol my child at a school in special measures, where they dutifully go from 9 to 3.30 with a lot of time off sick, but effectively home school them. During the swine flu thing we and our neighbours half seriously planned a school for in case there was a major closure; we'd have had profs or senior professionals for every subject, plus a booker prize winner (someone's father) running creative writing. All that, and the easier admission: great!
Proposals to tax private education fail, because it has no effect on its affordability to the genuinely affluent but cripples those who are struggling to do what they see as the best for their children. Systems in which children apply for some benefit (scholarships, extra tuition) fail because middle-class parents know how to game the system and dutifully fill the forms in. And so on, and so on.