OP I’d just say that if you aren’t someone who really loves maths, it can be difficult to understand how important it is for people who do!
My younger kid, who is good at maths but wouldn’t say he loves it, found KCLMS a bit stifling, especially socially - he’s pretty sociable and quite ‘normy’ (plays sports, has girlfriends, watches incomprehensible young men on YouTube talking about sharks) and KCLMS was an experience that he sort of put up with, for want of a better way of putting it. It’s called a ‘school’ for a reason; it’s quite strict (kids have to stay on site and work in free periods, no canteen, loads of work and very high standards.) He didn’t hate it, at all, and he wasn’t miserable there. But he (not me!) explicitly chose it over a local sixth form because he wanted a good shot at a top uni, and KCLMS gets brilliant A Level outcomes and uni offers. Without that factor he would have preferred a local college in a heartbeat and he freely says he would have had a better time there.
My older kid though sounds more like yours. He loves maths. For him, KCLMS was like heaven, and he’s still incredibly fond of it and keeps in touch with a lot of alumni and teachers.
I don’t know your son, obviously - but what I’m saying is that if he himself says he’s in love with maths, don’t underestimate the potential impact on him of being in a school where maths is explicitly prioritised and where he will be working alongside lots of really talented young mathmos and have real life mathmo mentors (and loads of curriculum enrichment). Unless the other schools you’re considering are extremely unusual, their maths provision will not come anywhere close. It absolutely changed my older son’s life, very much for the better. If you’re not mathsy yourself (and I’m not) it can be hard to understand how much this stuff matters to them!
Re private schools - I get the impression that private entrants were relatively rare, but I don’t know how far that was policy as opposed to not many private kids applying. As others have said the basic metric seemed to be ‘would this kid have the opportunity to get a really good maths education at their current/future A level school’. If you’re at a private school where A Level outcomes in maths aren’t particularly good, you might have a decent chance. (This is guesswork though!)