No, we shouldn't get rid of private schools. Most wealthy people have earned their wealth, and if they want to send their children to the best school they can afford that's their prerogative. It's not their fault so many state schools are of such a low standard.
I agree wholeheartedly with ckingalt. You shouldn't punish the fortunate for their good fortune, just try to spread it around more. I've heard a lot of people who say they hate that private schools exist but I bet they wouldn't say that if they could afford to go to them!
I've never been to a private school but I know people who have and it seems like they really try to ensure that every kid achieves their best, whereas at most of the schools I went to, the teachers just wanted everyone to get a pass at GCSE; not enough help for those aiming higher and not enough help for those 'anomalies' or 'special needs' who had been written off and were expected to miss the mark. If I could've afforded private education I would've gone, and if I can afford it for my kids I'll definitely send them there.
Also, I think most privately educated kids seem to appreciate their education more, because they know it's paid for. So when people talk about how their grades are 'paid for', I do think some merit should be awarded to the student, even if they had more help they still sat the exams / coursework themselves. If my parents had invested a lot of money in my education, I think that would encourage me further to strive for higher grades.