"Or is it that there isn't any angst around them, you put your child's name down at birth / age 5 / age 7 and if you are in the first 100 bingo you have the place?"
There's only one prep I can think of which requires a registration in the first month or so if the infants life. But there are various 'first come first served' preps where you probably need to do this is practice (because they start with nursery, and you need to put down just as early for them).
There aren't any truly non-selective private schools in London, and as long as you register before the deadline, your DC can be assessed. Some guillotine their lists at a certain number of applicants, so you dinMt want to leave it too late, but the school which did that the most (guillotine falling earlier and earlier each year) has now abandoned the system.
CLSG is anomalous, and has been much criticised (it's arguably a bepreach if ISC standards for admission). I think opting for this system reflects the ethos of the school, and should be counterpointed to nearly every other school - because they all face the same constraints and difficulties when making offers, but manage to have year groups with the right number of what they see as desirable pupils. Whether you want to deal with a school that honours its offers is a question of how a family views that ethos.
But I digress....
.... London state schools are generally pretty good, and a handful are very good. The snag is securing entry to them, and there are still large areas if London where you have no real choice (despite having 6 spaces on the pan-London form) and the only school you can be reasonable sure of getting an offer from might well be one you dislike (for whatever reason, and yes some (but not all) people will have shabby motivations).
OTOH, London private schools are excellent. The angst in threads is about people over-thinking very small differences between very good schools. And of course it seem a Big Deal right now as this is the peak of the private school admissions season.
There probably isn't much that doesn't happen in private schools that doesn't happen in state schools at all, but provision may be patchier in state schools. I'm thinking of range of sports on offer, level of facilities, abundance of supplies, better maintenance, amount of participation in competitions and enrichment events, co-curricular provision and university and career advice. Also, lack of (central and local) governmental diktat (which binds different types of maintained school to varying degrees, but which none are totally free of).
Whether you value those aspects of education is up to individuals. And of course has to be set against how you see the respective communities in the schools. Only a few are rich enough to have a bursary policy that really means a broader parental income spread, but not everyone is incredibly rich (especially once they've shelled out for fees). They are probably more diverse than they are sometimes given credit for - especially those which attract the offspring of the international business community.