I guess it depends on how you define "quite rich". The reality is, private education is objectively beyond the reach of most families in this country. Most families simply don't have that level of disposable income going spare, no matter how hard they try to cut down on other non-essential expenditure. And yes, there is a tiny number of full bursaries available for those on the lowest incomes, but the reality is that they are few and far between.
I do understand the reference to nursery costs as the cost of a full time nursery place is often broadly comparable to private school fees, but I don't think it's really the same. Firstly, families on all but the highest incomes get some help with nursery costs, so won't usually be paying the full whack. Secondly, kids only need nursery care for a few short years, and lots of people rely on savings or loans to pay for it, rather than funding it from income alone. And even then, lots of people report finding it a massive struggle, so I think you're wrong to assume that everyone who pays for full time nursery could just carry on paying private school fees until the child is 18.
I'm not speaking from a place of resentment. We could have afforded private education ourselves and would have done if we had felt that it was worth the investment. But I don't think we are typical and I don't think that anyone who isn't "quite rich" could even contemplate it. I get that the word "rich" means different things to different people, though, so replace it with affluent, comfortable or whatever term you prefer.
I still don't understand why it really matters, at a societal level, if some middle class kids go to state schools instead of private schools. It has always been the case that private school is beyond the reach of those who can't afford it, and that won't change... it's simply that a few more people will be tipped into the "can't afford" category, and while that's a shame for them, I'm not getting why it matters for our society as a whole?
I'm not being bolshy here, I am genuinely trying to understand your point of view. Of course, you're right that this policy won't affect the super rich or make them care, but neither does the status quo - we need other policies to tackle the elitism in our society. In the meantime, if this policy raises some extra revenue for state education by taxing something as a luxury which, in my view, should be treated as a luxury, that seems pretty reasonable to me. I don't really understand why the argument that it won't affect the super rich means that it isn't worth bothering with it at all.
As I've said above, I do have sympathy with those who have opted out of state education because of SEN that aren't being properly met, and I think we should seek ways of mitigating the impact on this policy on those families.