The point is that Oxbridge do not accept the best people - the best people have not had the opportunity to prove themselves to be the best - instead they accept the people who have proven themselves to be the best. This is disadvantageous to those who have not the same means to prove themselves, and the fault of this lies in the public/private divide.
PS_ as for the highly performing state schools, they dont really count becasue parents move to be in the catchment area, the better the school the higher the surrounding property value, the average wealth of the families in the surrounding areas goes up, so not only are they getting a good education, they're rich which sounds surprisingly familiar... oh yes, private schoolers!