Why the process is not fixed from that part? Redesign in a way that private schools will not get any benefit from the process itself.

How exactly? the way it's managed at the moment is that the admissions tutors keep an eye on what different schools are doing in terms of prepping their students and keep that in mind when assessing them. Which is exactly what causes the 'bias'. I am significantly more impressed by someone who manages to get good grades with little help than someone who manages to get excellent grades with the most effective teaching of the test that money can buy (having gone to an independent school, I was one of those being 'discriminated' against, but I can well understand that many of my classmates, who primarily good at cramming for the exam and not understanding the material, got very good grades but would not do nearly as well at university. Heck, I got in and struggled way more than I had done before, and more so than many in my year who had worse grades in less good schools)

(I would, in general, be in favor of fixing GCSEs and A-levels. They have persistently moved in a direction that rewards memorisation of particular keywords, something which especially rewards teaching the test, as well as getting easier and especially less good at discriminating the top end of ability accurately. But it's still not going to be enough to remove this difference)

It is, via the interview. This is why A-level results are a coarse filter, and why they have different standards for state vs private schools; state school kids with 3 A's presumably excel in the interview to the same extent as private school kids with 3 A*'s.

It cannot be understated how much of an advantage someone who went to a private school has over someone from a state school, with respect to the entire process (exams/admissions tests/interview prep).

What is your specific proposal? Anything you do, the private schools will do their best to adapt.