Private schools (and indeed parental choice in general) may seem like a good idea in principle, but the unintended consequences in the long term are the polarisation of society and an increase in inequality.
So what is needed is a way to reintegrate private with state, and therefore the pressure on private schools to work more closely with their local community is a move in the right direction.
[A more interesting question would be what would need to happen before schools, that are currently funded exclusively by parents, became funded by the state]