If top universities selected only by grades, I think you would still end up with a high proportion from independent schools. It is not like a random 7% of the population attend independent schools. The top schools may be selective, only letting in students with better grades, or may have scholarships for bright (or athletic) students with lesser means which push up their averages. Those students who pay will generally have had an early childhood where they got good nutrition and plenty of attention and nurturing. If they were falling behind, their parents would more likely be better able to pay for extra tutoring. If the schools has more bright students than average, there can be positive feedback effects. There are also subjects which are taught more at independent schools like classics. It is hard for a university to find a lot of state school applicants with prior education in classics (the solution is to start with students who did well at history and give them a year of preliminary studies before moving to classics proper.)

You see the same effect with grammar schools (in the U.K.) or magnet high schools (in the Us) which feels a little more egalitarian, though of course the availability of such schools can vary a lot geographically (and a higher cost of living near good schools can push out poorer parents too.)

I don’t really know what the solution is. It seems unmeritocratic to expect universities to take only 7% independent school students (for one thing I would expect they make up slightly more than 7% of the university student body; for another this would probably have the undesired effect of having the 7% at Oxford and Cambridge mostly made up of the top public schools) but it also feels slightly unfair to only look at grades (indeed the universities do try to be more accessible and take into account the fact that some students have depressed grades due to poorer schools)