They can draft VAT regs fairly easily to include some but not all of the activities of private schools e.g. to include education but not wraparound care, and to exclude say catered meals. Likewise they can ensure HE services are exclued. HMRC do know how to draft VAT regs, they have some experience in it. Its not as if education is some highly specialist legal term which only admits of one meaning so if they apply VAT to some education services automatically everything else MUST be included.
It will no doubt still end up in tribunal and there will be weird edge cases because that's how VAT regs work - jaffa cakes are cakes not biscuits, a poppadum is a food not a snack because you're meant to put chutney on it before transferring it to your mouth etc. You'll get pre-preps arguing we only do formal ed for 5hrs aday and the other 5 hours kids are on the premises were doing VAT exempt childcare so the VAT can only apply to 50% of the fees. That's what I'd do if I was advising schools on how to minimise their VAT liability.
But the reason they are going for this not the charity status thing is they already did that with Charity Act 2003 when it was assumed the codified public benefit test in that would exclude most private schools from operating as charities. Instead the ISC v charity commissioners judgment in 2011 found in favour of a really wide interpretation of public benefit enabling a lot of schools to continue to operate as charities and enjoy those tax advantages.
It seems like most indy school parents don't know this but the way private schools operate in this country to purchase education for a really small sector of society (about 6%) is actually pretty much uniformly opposed on the left. All Labour governments want to get rid of it and always have. There are other countries which have more diverse indy school sectors with more like 20-30% of children attending, and parents paying top-up fees - Australia is like this for example. But these indy schools are usually religious and the top up fees are v low so accessible to a larger cross-section of society meaning the function of providing private schooling and even subsidising it is to enable access to education that respects religious values - not just bakes in social inequality. Our income based segregation within schooling is a huge driver of social inequality so the eventual closure of all private schools would be seen as a positive outcome for most on the left.
This is why all the arguments that this won't raise that much VAT etc because parents will remove their children are irrelevant. Parents removing their children is the point, on the left people want parents to place their children in state schools.
Private schools have huge advantages to staying open though, including that many but not all have charitable status giving them tax advantages that state schools do not enjoy. They also cannot simply be brought into public ownership - because of boring things like rights to private property under the ECHR and historically the right to run a business protected by the EU (which obviously no longer applies and has no parallel in English law). you will meet hardcore lefties who want to nationalise public schools - which is effectively what happened in the 60s when endowed grammars had the choice to become comps or become private in most LEAs. But legally it would be fiendishly difficult.
VAT however is a really easy win. No court can argue that the govt has the authority to decide what goods and services fall within the scope of VAT regs - that's clearly a policy matter. Leaving the role of the courts to decide the edge cases but not undermine the actual scope of the policy entirely.
If you understand that reducing if not ending private education is the unspoken aim of this policy its going to work perfectly well.