There is absolutely no question that the attainment gap would be smaller if private schools didn't exist.
Look at Finland, for example, which abolished its fee-paying schools and instituted a nationwide comprehensive system from the early 1970s onwards.
Not only did such reforms lead to the closing of the attainment gap between the richest and poorest students, it also turned Finland into one of the global educational success stories of the modern era.
I can't see the abolition of private schools happening here (yet, though I do think it will come) because it's a mass middle-class vote loser. But I think the widening gap - and the awareness of it - is making the status quo re private schools unsustainable.
I think (hope) they will be integrated within my lifetime.
I used to teach in a private secondary and I thought it was an awful environment for kids - appallingly pushy entitled parents (mostly), hideously unhealthily competitive, utterly obsessed with results, and not at all conducive to nurturing a love of lifelong learning.
I know it's a generalisation but it was my experience.