Free Education is a right for all and legally has been in the UK since 1944. This right was what promoted state schools to be set up all over the country catering for all children up until they are 16. Private education existed before this and exists alongside it. Anyone who can afford it has the right to use it if they so desire, but there is no right for someone to be provided with it if they can't afford it.
Private Education always was and still is a luxury for the rich. There is no obligation for anyone to use it and no one has a right to use it. As such having VAT on it was fine, because it is an unnecessary luxury product.
What's muddied the waters, is the huge and awful numbers of children who now end up in private school because of the lack of provision which meets their needs in mainstream. This has fundamentally changed the whole role of private education, because it has now meant that significant numbers of children are using it out of necessity, because it is the only way they will get an education. In looking at the VAT situation, it means that now families of children who cannot be educated in other ways are paying VAT on an essential product.