The thing is, it isn’t always an sducational advantage.
I have just compared points at A level / equivalents across the secondary schools in a town with generally good schooling.
A state and a private school share the top score - both schools are selective.
Then a private, about 5 points lower.
5 further points lower, both still well above national average, come a state and a private, both non selective.
A non-selective state cones next, then another non selective private a couple of points lower.
A final state, in an area of high deprivation, brings up the rear.
It is a picture in which the answer to ‘is private or state educationally better?’ is ‘it depends’. The best non-selective state is better than all but the most selective 2 private schools.