Briefly on the "white middle class" thing - my husband are practising Catholics living in one of the whitest parts of the country (rural Cotswolds) and we have been to 5 or 6 Catholic Churches as well as Clifton Cathedral, and let me tell you it's the most diverse group of people I've honestly seen anywhere in church or in the rural Cotswolds.
When I went to C of E churches it was literally all white even in larger towns and cities... in the Catholic Churches there are black people, south Asian people, East Asian people, South American people, different race couples with mixed race kids - anyone you could imagine in any combination, they're there... and most of them are super hardcore Catholics too (which I adore! It inspires me), more so than many of the white people - and nearly all of the non white attendees are 30s to 40s couples with kids.
The only demographic that is significantly whiter is the much older people (like 70s+) but even then there are exceptions - I can remember one specific south Asian older mother figure with her daughter or maybe grand daughter at mass, the older lady was in a beautiful soft rose coloured sari with gold detailing, which is why I remember her!
We have been to the churches that the catholic schools go to and from what I remember there was a fair mix of ethnicities there too within the school kids - despite being the whitest part of the white as anything Cotswolds 😂 but it was less diverse than other churches that we had been to in other areas for sure.
Furthermore, I think that Catholic schools are only allowed to select Catholics up to 50% of the student population, and then it's a free for all, Catholic or not - but the parents do have to accept that their child will be going to mass etc regardless of the home faith (or lack of one) so if they don't want that they shouldn't apply. I don't think it's wrong that faith schools exist (as I think many some people here might be implying but I could be misreading) and I think they should prioritise those with the faith, for obvious reasons. I just think that all schools, grammar, comp, etc all need massive improvement. I went to an all girls grammar and then switched to a co-ed comp and both were essentially hell on earth. 2 out of 4 of my primary schools were also pretty bad. I also worked at a private school and from what I could tell it was alright for most of the girls there. All of this was in the Cotswolds and there was always a mixture of ethnicities (to varying degrees - the private school was the most diverse) which, if I were to think about it in a purely analytical way, were probably over represented for the actually makeup of the local community - but that obviously wasn't an issue for anyone.
So I guess my point is that it's not always what you might think, and in my experience of all different types of school and my experience of specifically Catholic Churches, even in the rural Cotswolds, there never seems to be race bias or filtering. Obviously it could be different for different parts of the country, but that's just my personal experience. I obviously can't speak for any other type of faith school as I have no experience of them.