Is that the same university that admits more from eight private schools than from 3/4 of the state school system? Yep, sounds like a fair system. Cool story bro.
1. Differences in encouragement from parents, teachers etc. Differences in how much emphasis is put on education.
2. Related to #1 - state school kids may think they are not good enough and not apply.
3. Actual quality of education may be significantly better at those schools, especially in certain subjects.
4. Genetic potential - some traits that allow kids to do well academically may well be genetic, so it could be that the parents would also have those traits, be better off economically because of it, and send their kids to good schools because of the combination of valuing education and being able to afford it.
The point is that:
1. The gap is very large. Those schools are only graduating a few thousand compared to hundreds of thousands students in state schools. Even if you compare with institutions that receive a huge number of applications from private schools (e.g. Edinburgh, Durham), the effect is disproportionate.
2. We need to make use of people who have the most ability. It is foolish in the extreme to suppose that going to Eton or having the right parents makes you a capable adult. You say the admissions process isn't biased...but we are optimising for whether you attended Eton. Speak to someone who didn't attend Eton or similar, and ask them if you think that is fair.
Btw, I went to a private school and went to a good university (that takes from private schools to a very high degree), and worked as an equity analyst (so meeting C-level execs and fund managers who mostly went to Oxbridge) your views are just generally very very wrong. These people are not capable or unusually smart. They just grew up in a safe environment with class sizes of less than 30 people. And UK management teams are, comparatively, very weak because there is no real meritocracy here. Sure, there are some good managers but this is basically certain in a large enough sample. And if someone has come up the hard way, they are usually a much safer bet. It is kind of crazy that you have picked one of the most busted, unmeritocratic societies in the world...and tried to show how it is actually fair...
To back this up, nobody else at my college applied. After I did on a whim, I discovered there was an aptitude test to pass, which I didn't. I blame my negligence for failing but I imagine people from more prestigious colleges & sixth forms would have been aware earlier and had help studying for it. Or even had private tutoring specifically for it.