(Yes, there are other exceptional universities - Oxbridge used here as shorthand)
The issue is not about Oxbridge not giving first dibs on places to private students, it's about Oxbridge trying to reduce private numbers based on a totally fallacious percentage.
You need to drill down and look at the number of students who achieve appropriate standards for Oxbridge then split that number into private and not.
A privately educated child should have no more and no less chance of getting to Oxbridge than a state educated child of the same ability. Plus selective state schools should be included in this in some way other than as state - they are pickier about their intake than a lot of the top private schools.
The top private/selective state schools have already filtered out the "non-Oxbridge" candidates before they start school where non selective schools can't.
(Ignoring selective states now for ease)
You can allow for private school advantages by requiring reasonably higher grades from the private schools (but not crazy) to assess admissions standards.
VERY VERY simple example, (numbers chosen for clarity not because I think they are correct), but they show the effect of this filtering.
Imagine the following numbers:
Private Schools between them have 1000 pupils, 50% - 500 of whom reach the standard chosen for private schools.
State schools have 15,000 pupils, 10% - 1500 reach the standard for state schools.
If you give places to all the 2000 I've shown, you've got 25% (500/2000) from private school, vs the 6% (1,000/16,000) you should get if you go by number educated privately.
Or you could just take your 6% (60) of the private pupils, leaving you with space for 440 more state pupils who didn't make the grade - but then you as a university are missing out on 440 extremely talented students, and taking students who may well struggle with the curriculum.
Doing it with approx. 20% private at A level and same proportions of ability you get:
Private Schools 3000, 1500 at standard.
State schools 13,000, 1300 at standard.
Then you have 2800 students of the admissions standard -- 54% (1500/2800) who are privately educated! If you've then only got 2000 places you could give the first 1300 to the state students - but that's then purely social engineering (not discussing the rights and wrongs of that here). Making the private school requirements so high as to reduce the numbers to the "right" proportions would be the same thing - you can't realistically assume that a private A star is only worth a state B given the private students are already filtered by ability. Or you just say your intake should reflect the 54% that get the relevant standards from each sector.
The government/universities need to decide how to measure ability then aim for those percentages - or decide that social engineering is more important.