The privately educated still contribute in taxes towards the state education of everyone else, in addition to any fees they pay for private education. Closing private schools would therefore simply make all the state schools worse off, as there'd be an extra burden in terms of pupils on state schools.
Private schools tend to be academically rigorous institutions. As many people have said, lower-achieving state schools should try to emulate them in terms of academic achievement.
Also, getting rid of private schools would not solve any sort of problem in terms of inequality of education. As long as people have money they will treat education as a commodity, and it cannot therefore ever be 100% fair. If you got rid of private schools, you would have a lot of rich parents who would simply send their tutored-up children to the highest performing state-schools, thus increasing the competition for places and, again, all you're doing is making those who are clever but can't pay for education inevitably lose out.
I would suggest we keep the private schools while creating a tiered State system with grammar schools. This would be the best way towards increased social mobility. Sure, it's not perfect, but it seems the fairest of the options that there are.