In answer to the OP, I'm not sure 'abolishing' private schools is the answer, but making 'contextual' university offers might be, which is sort of already happening.
The response of the wealthy who could have expected an Oxbridge place until recently is Ivy League instead, given that the fees are increasingly on par.
But the charity status has absolutely got to be ended.
What this country desperately needs is a New Deal. Millions invested in state education, teacher training, schools, tech colleges, quality university degrees. And in return we have to demand better from our state education and from our DC.
So many of us have accepted mediocrity for far too long, we are proud of our boorish ignorance, our xenophobia, our small ambition; we nurture our hatred of 'liberal elites'. All the stuff that brought us Brexit.
Meanwhile, the 'liberal elite' are running further and further ahead of the rest of us, in some cases having resigned themselves to Brexit, they're now looking to advantage themselves via the forthcoming bonfire of workers protection, safe in the knowledge it'll be their properly educated young who will hold the whip hand.
We could do worse than emulate Germany.