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Private schools don't need to be scrapped
Their charitable status needs to be scrapped, however, and their licence needs to include an obligation for significant numbers of affordable places.
While we're at it - selective schools should have either state funding removed entirely, or intake criteria to have a minimum of 50% nonselective (other than those employed by nonselective schools).
[Edit] so I'm a day late with all of this, it appears.
Private schools don't need to be scrapped. There will always be a place for peolple who want to pay for education. The old boys / girls network fast track is one thing that needs fixing. But the main thing is funding normal schools and rewarding teachers for the job they do.