What's more, the people involved in the application process know the reputation of schools as well, both in respect to what kind and how much bias they have in terms predicted grades and in terms of how much the school tends to 'teach the test' or otherwise be good at grooming their students to get good grades or perform well at the interview, so a student who does well in grades or interview from an underfunded state school is more likely to get an offer than a pupil with equivilent results from a high-performing public school

There's still a substantial bias in offers towards public schools, but a lot of that is because such schools are selective like the universities, actual student ability does still matter and better schools still improve student ability more, and there's a strong self-selection of candidates for top universities (a lot of people who might get accepted for oxbridge don't apply because they don't think they have a chance) as opposed to old-school elitism in the application process. (Not that all of the above factors are actually fair from a grand socioeconomic viewpoint, but they are reasonably fair in as much as the university judging the merits of the applicants it gets).