Not at all. I've known plasterers and and a small shop manager send their child to a private school when they've been bullied or led astray in a crap comp. They can't really afford to, but can't afford not to.
Lots of private schools aren't for the Russian oligarchs - especially out in the regions. Our nearest (the only one in our area) is a kind of specialist school for kids with "troubles" such as bullying, gangs etc., and is a chance for them to walk away from a bad environment to turn their lives around -it's not particularly academic, more of a "safe haven" where kids who want to achieve can flourish in a safer environment but only if they really want to! By all accounts, it's remarkably good at doing it and charges nowhere near the kind of charges that are banded about social media - it's expensive but not tens of thousands per term.
Then we have all the genuinely "special needs" schools for various mental and physical disabilities where the local authorities can't provide the care and education that they need and deserve. Again, "normal" parents scrimp and save to put their kids into such schools.
We really need to challenge the narrative that private schools are only for the mega rich. Maybe that's the case in London/SE with wealthy foreigners etc., but really isn't the case out in the regions.