These threads always seem to assume private/independent schools are all £20k/£30k elitist institutions, or are London super selectives with parents to match, lots if international students etc etc. Of course some are, but I would suggest the majority are £10k/£15k small local town indies, who parents are actually middle earners, could have gone to local state, but chose, for whatever reason, to send their DC's to private. They tend to be in nicer areas, and the local comp will also not benefit from contextualised, their DC's will likely have extra tutoring, because the same parents, doing the same kind of jobs, live in the same houses, in the same postcodes, and have the same wealth and experience. They are all therefore equally privileged.
These are the DC's who will be squeezed out of Oxbridge. They all end up in other v.good Unis, they have to go somewhere, right, who will then be accused of not being socially diverse enough ie: Durham, Imperial, St Andrews, Bristol, UCL, LSE, Exeter, Bath.