We shouldn't ban private schools because that would be immoral. The only justification from banning something is if it causes people harm (of course, not everything that can cause harm should be banned, but this is a necessary condition). Private schools do not harm those who go to state schools - they provide an advantage to a certain group of people above others. To say that 'we should ban things that provide people with an unfair advantage' is to allow some very totalitarian policies. Using the same logic in other fields, we could justify banning just about everything.
The only solution is to make state schools as good as private schools, although I highly doubt this will ever happen.
Besides, when people point out that private schools and Oxbridge dominate, they seem to ignore the fact that people attending these institutions are generally more capable. Given that all private schools are selective (either socially, academically, or both) it is no surprise that they produce the best pupils. They take in a much more able intake, and then provide a higher quality of education (in general). It should come as no surprise that their alumni are so successful.