I am not following. Our kids were educated privately because we lived in an area where inner city state provision was very challenged. Most people we knew at NCT stage moved. Of those that did not, the largest number played the system by either renting in other catchments (very common), by tutoring for grammars or specialist music places, or by finding religion.
None of those appealed. DC did try for selective state places but were not offered places. Paying for education was the alternative, and was a real struggle but we don't regret it. Yes, of those I knew locally more from state schools got into Oxbridge, but so what. (The variation tended to be larger so there was a correspondingly larger number who struggled.) Oxbridge is not the be all and end all. Education matters. If we had lived elsewhere in the country it is very unlikely they would have gone to private schools, but we didn't/couldn't. If you look at the stats, private education is much more common in London.
I think part of the underlying debate is what people see education as. Is it grades and getting into Oxbridge, or is it broader. The most obvious difference between academic London private school and very selective London grammars is not academic or results, but access to broader education. Things like sport which help develop resilience and drive, as do drama and art.
FWIW US Universities seem willing to offer good scholarships to those who are exceptional regardless of the school type.