I haven't read all the replies, so apologies if I'm repeating some of the points.
Firstly, I wouldn't select a school on the basis of university (for all you know your child may not want to attend university after all).
Secondly, I do believe those children at private schools have far greater opportunities and are at an advantage. Yes, they will work hard to get the grades, but remember they have far more teacher to pupil time, a lot more support and generally come from comfortable homes (easy access to laptops/wifi etc).
There is a drive to help encourage those from disadvantage backgrounds to apply to oxbridge, and there should be. There is also a drive to help children from ethnicity minorities to apply. This doesn't take away the credit for the hard work of someone at a private school, it is merely a scheme to try and encourage and allow access to these universities.
Remember student from 'ok' local comp, achieving A A A at a level, is generally considered a better candidate than student from private school with the same grades. This is because student 1 had far more obstacles, had to garner the support themselves (usually) and quite possibly is the first from their family to even consider university.
Student from private school will have teachers supporting and training students for interviews/exams. They will have parents who are generally doing well and can afford more help (outside school tuition). That is why the difference - which has to be there to allow these opportunities.