I find this a really tricky one. I kind of argue back and forth with myself on it!
Firstly, to answer the OP’s questions, I always knew about (I) but only found out about (2) reading threads on here.
But you find that, ultimately, the many threads on here aren’t really about VAT but our extreme two tiered education system in the UK and the privilege a private school education confers. This does seem unique to the UK. Yes, other countries have private schools but there isn’t the same pride or advantage in sending your child there that many have in the UK. It is just another school and another choice. And, generally (yes there are exceptions, especially in Switzerland and the U.S) the fees aren’t comparable to those in the UK.
We came fairly recently out of an old money system where your parentage and breeding determined your chances in society. We can now relatively easily buy privilege and private schools offer that. There are many people who might have been termed ‘nouveaux riches’ slightly haughtily who choose private schools, so that disappears within a generation (which is probably a good thing).
But, ultimately, if you think private schools are a societal problem, you should be brave and abolish them, not just make them a bit more expensive.
However, I do think that those that are managed well and understand they have many important stakeholders, including the community in which they are situated, can be a real force for good (Westminster, which intends to go ‘needs blind’ in about a decade, and Charterhouse, which runs an amazing hub for aspiring Physics teachers, spring to mind).
I, personally, would be in favour of mandating private schools to fulfill their societal obligations (certain % of full bursaries, sharing of facilities fully when not in use, not just occasionally, supporting state pupils in rare sixth form subjects). Some schools are already meeting those criteria. Others would have to make large adaptations and maybe increase fees to do so.
But, with the above, it would make a clear positive difference to the communities and those who chose private would maybe embrace the communitarian aspect as well.
VAT on the other hand won’t raise a lot and will be hard to introduce compassionately.