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Sure thing, but there's quite a big difference in income/status between someone that can do local private school fees of £17k and fees for places like Eton which are £50+k per year.
I think what I'd do as Starmer is to take an approach something like universities. Regulate the term fees to make them more accessible and perhaps mandating wider entry conditions/ free places in return for the continuing tax breaks.
I love you man but if a family can afford an average £17k per year toward private education for their kids, they ARE especially wealthy. They may be making other sacrifices, and I'm not downplaying that. But if you've got £17k a year to blow, you're doing alright.