It's a difficult one, because as much as it is easy for someone like me to say they should all be closed, I know for a fact that if I had the income and I had a kid I would be sending them to one.
It's also too easy to say that state schools should just be made as competitive. You're living in a dream land if you think that the public purse can pay for that.
Personally, I would have fewer issues with them if it wasn't for the concept of nepotism (and that probably only really exists at the top end of independents anyway - Eton, Harrow, etc). I don't think we as a society can sit back and say "all's fair" when we have a cabinet, top jobs, etc, dominated by independently educated individuals. If a true meritocracy exists, and I believe we should be striving for that, then you would expect a much lower proportion of independently educated individuals in the upper echelons of influence - of course proportionally they will be larger than 10%, because of their superior education, but not by the amount seen today.
Will eradicating private schools achieve this 'true' meritocracy? Probably not. It's difficult to say that it wouldn't shift a focus in education and opportunities, though.
I think the solution lies in a gradual shift to making independent schools centres of academic excellence (rather than opulence) by ensuring a large number of poorer individuals can enter through substantial bursaries. If 50% of your students came from poorer backgrounds, you would get a mixture of cultures and ideas. Privileged kids would have to mix with poorer kids, and that, in my opinion, would be of benefit to everybody - especially if those pupils are to be leaders of the future.
So yeah, meritocracy.