I see the logic in VAT on school fees. Sounds fair enough in theory, but in practise, I don't think it would make education more equitable at all. Probably it would have the opposite effect in reality.
There is already a huge range in 'private school fees.' I've heard, in certain parts of the U.K., it can be £3k per term? Where I live, it can be anything from £5k per term, to 8-9k per term for the 'bigger name' London Day Schools.
At a school one of mine went to in London, fees increased from 7k ish to 9k in the time she was there. But the school has a significant bursary programme - 25% receive some kind of bursary, of which about half are on full bursaries. A proportion of the high fees people pay funds the bursary programme. In addition, local schools use the sports facilities regularly. At weekends local families can use the pool etc. Following the Grenfell fire, the school absorbed the students and teachers from an affected school so they could finish their GCSEs. The school won a TES award for improving social mobility (alongside the NHS and the Home Office).
Its one thing if you live in what could be called 'middle England' and there is probably not much difference between the local comp and the local independent. It's very different though in places like London where extreme poverty and extreme wealth exist side by side. One of mine (for SEN reasons) attended a local state school - there was a fatal stabbing in broad daylight on school premises, witnessed by students.
Many of the independent schools around here are not 'accessible' anyway, regardless of how much money you have, because they only take the top 2% in terms of academic ability. These schools are ridiculously competitive in terms of 11 plus entry. Children of politicians, celebs, whoever, are turned down more often than not. Doesn't matter who you are if you're not in the top x% in the entrance exams. And if you perform very well, but can't pay the fees, they will want you. They want the most able kids who will get the results to maintain the school's position in the league tables. Even so-called 'back up' independent schools in London are still likely to be 4 or 5 applicants per place and entry is via academic selection. There is no such thing as 'pay your money and waltz on in.'
All that will happen if VAT is put on school fees (round here at least) is that bursary programmes will be hit first. Schools will adjust fees because they are businesses. Some families won't care; some will just pay anyway and those who can't will move to where there are safe state schools with no stabbings or other violence. People who attend good safe state schools around here already pay the equivalent of school fees anyway in inflated house prices and stamp duty to move into certain catchment areas.
Its a very complex issue and the reason people use state schools is very region-specific. If we lived in an area with safe state schools we would use them. DH and I are both state-educated and of course we know it's absolutely fine, even excellent, in many cases. But not always.