OP, it's obvious from your posts that you're not British.
It's possible you're deliberately stirring up existing UK social division... but maybe there's just a bit of a mismatch because you're missing some UK cultural perspective.
So on the chance that you actually do have a 10 year old at a private school you're not happy with and you're disillusioned with the whole promise of a UK future, here's the UK take you might be missing:
1.A very British-valued way to help your DC is by supporting them to genuinely become the best possible version of themselves. That starts with your parenting. Treating their education as a paid road to grades will help them less than supporting them from wherever they are as an individual to achieve the best they possibly can in a direction that's right for them. The UK preference is to do that through support rather than shaming - we think it develops a better character.
2.You really need to put your own personal effort into their education, not just money. You've chosen their private school badly - which suggests you didn't put in sufficient research and effort when choosing it (eg moving if necessary, tutoring and support for selective entrance, identifying specialisms like music in schools which would boost their application). If you switch to either another private school or a state school with as little thought, you will likely be just as disappointed
3.You also need to put in effort whilst they're there: taking an interest in their education and their friendships, encouraging them to make good choices and follow up opportunities. This is parenting. No school will fully do that for you: although some will make it easier than others.
4.If your DC is 10, then they are presumably about to go into year 6. This gives you the opportunity to change for secondary at the natural transition point. So your complaint that the schools are full makes no sense.
5.You just about still have time to make a difference to which state school they have access to (although you should have started looking around 2 years ago. Yes, it's a hassle. That's what I mean by putting in effort.)
UK state schools vary hugely - much more than the difference between private and state. But being at a state school certainly won't block them from any future career, and may even make access easier
a)You can still move before the application deadline to get within distance of a better school, although you will have to make it a priority and move fast
b)You're too late to take the religious route now.
c)there's still time for some last-minute tutoring for selective entrance, but it will have to be intensive and focused on technique for the specific exams
d)probably too late for school specialisms like music, although you should do the research to find possible schools if eg your DC already does music or sport at a high level
6.If your DC is actually about to go into year7 (birthday this month) then it's too late for normal application for state, but you can still go on waiting lists. At this point in the year, it doesn't matter that you didn't apply in time, state school spaces are given on proximity regardless of when you joined the waiting list. There's still often movement even into the first few weeks of term as people shift around. It's not as easy as having the space from the start, but far from a disaster - they'll be there for 5-7 years. Again - this is part of putting in the effort and being flexible.
7.If you would rather stay private, then you could certainly find a great school to move to. The big thing you buy with private school is wider choice - of course, still within constraints. Make use of that freedom! It does still take effort and will. You may (like with state selective) have to do some fast tutoring. You may need to move house (but there's less pressure on home location for private).
8.You can also supplement their education yourself,through tutoring and most importantly through your own influence and engagement. Regardless of sector or school.
9.You can't buy access to a particular set of people. Don't make that part of how you select a school.
10.The right private school can genuinely teach personal skills and strengths that your DC will benefit from throughout their lives. But it's not the only way to get those skills and strengths. The most important thing is your personal engagement: both in choosing the environment (which requires your effort and judgement regardless of sector) and also your support as a parent
If your concern extends to university, and to the suitability of the UK as a place for your DC to make their life, then you should certainly think carefully about what you think is best for your family. You don't need to make sweeping condemnations of UK private education in order to justify a choice for your family which you actively and carefully make.