I don't think that the data supports your position. You appear at the outset to have mistaken the point that I was making, which was confined to competitive advantage on admissions, and not about some wider advantage, whether educational, social, or otherwise. The numbers suggest that the number of privately educated students who obtain places at Oxford and Cambridge is disproportionate to the relative populations of private and state schools. It appears that educational opportunities, and corresponding social mobility, are still to some extent impacted by the division between state and private schools, although not to the extent that they were, say, fifty years ago. There's another, less easily measurable division between good state schools in affluent areas which attract the best teachers and the most committed students, and are supported by parent groups, and state schools in less affluent areas which may struggle with staffing, and with student and parental engagement.
Contrast another measure, ethnic diversity. Oxford and Cambridge admit ethnic minorities in a proportion slightly higher than the relative proportion in the population aged 18 to 25. Success in widening access by that measure has not thus far been matched quite as well by widening access measured according to socio-economic class. The student groups at many of the most popular universities remain largely middle class.
Media articles can only go so far, and may be spun in various ways. I was interested to hear my friend, a thoughtful woman who doesn't leap to conclusions, express concerns based on her first hand experience of how much more socially divided her university appears to be now, as compared to when she was a student there long ago. ISTR that you mentioned somewhere that you decided when young to say no to an offer from Oxford because you had concerns about the behaviour of some of the students there (although perhaps those concerns lessened later on, as IIRC you mentioned somewhere that you studied for a PGCE in Oxford). Where did you study as an undergraduate?