OP, I think you've got it spot on here: [white parents] ...don't even see when they are moving themselves into the centre. Because they might generally feel safe, and/or they haven't felt discrimination or prejudice meaningfully and negatively impact their lives they don't see that people who do experience this pretty much every day of their lives might want a break from it for themselves and/or their kids.
That's my experience of being white and middle class. It's also my experience, as a woman, of the perspective men take when excluded from women only spaces or events (and it has to be excluded, not discouraged, because many don't take hints. Certainly not 'encouraged to think about someone else' because they won't). Except that my experience is that men are arsier, more argumentative and demanding about it. All that 'why can't I access this women-only space? Why am I not welcome? That's not fair! I feel discriminated against! Where is 'men's hour'? Who are these delicate flowers who need their own time and space? They need to learn to live in an equal society (and be grateful for it). My taxes are paying for this. I have rights. I demand my equal right to access! (Some of the more unpleasant ones will make a nuisance of themselves to persue this point). Plenty of women take a similarly sneery but less aggressive line on women-only spaces.
As a white parent of a white child, who doesn't know much about ballet but who sends a child to ballet classes, I didn't even realise, until I thought about it, that ballet was a 'white art form'. That is, of course it is historically, because of the countries it originated in. Russians, in Moscow and St Petersburg anyway, have mostly been white. Parisians too, until more recently.
Now though and as a children's hobby? It only really struck me when I went to dd's dance school show and almost every child looked the same; white with long swishy hair. That represents a large sub-set of the neighbourhood but certainly a specific, unrepresentative sub-set.
The examples that jumped to mind immediately on reading your OP were of publicly-funded services aimed at low-income families, which MC parents will happily swoop in and take, if it's convenient to them. Parents who could easily have paid for private baby yoga / massage / signing lessons, or whatever, without even noticing the expense. Without thinking about why those classes were being offered free, by a children's centre (or why those exist), in those neighbourhoods. If they did, they'd just think the target population should have been quicker off the mark, weren't organised enough to benefit from them, or maybe didn't want them.
Would I send my dd to the ballet school you mention? I don't think so. I hope not. But I am assuming there are other ballet schools nearby, also within walking distance, that don't have prohibitive waiting lists. Would I inconvenience myself to go to a more distant school, if yours was my 'local' and had spaces? Maybe a bit, maybe not hugely (i'd probably decide against ballet and look for another local activity).
Whereas, if it was plainly a 'specialist academy', 'welcoming BAME children', with nothing about welcoming other member of the local community, then maybe I'd know not to ask.
I think there is a small ambiguity between 'encouraging BAME children within the school' and 'encouraging BAME children through the existence of the school and in all that it does'.
And I think you have to treat 'the privileged' in whatever case, as being a bit thick. State the rules clearly. Really spell things out for us, slooowly.