That is a concern of mine too. It's a difficult balancing act but at the moment I think poorer people are disproportionately affected by an opt-out private system for the richest families.
I think the education system is due a complete overhaul. In an ideal world will evolve in the digital age away from the industrial revolution factory model. We put children in, we get workers out - on the way we perpetually kill off the curiosity and playfulness of some of our brightest humans. The need to remember dates and facts and do other such performative skills is rapidly becoming obsolete in the digital world.
I suspect the answer is to be found in capitalism and Government socialism combined. Something that takes the competitive standard raising of private business but the inclusionary ethos of a more socialist Governments. I don't know what the answer is in the long term, but a short term fix is bringing the money from private schools into the public purse and redistributing it amongst all students.