I've been pondering what, if anything to say on the thread as I have a dc at KCL, but its always difficult to talk about these things without accidentally coming across as an arse.
For my dc KCL has been the right school. Being amongst many other gifted mathematicians means they are thriving. They are much happier than they were at their old school because they now have peers of a similar ability. It's not just from an academic point of view that they are thriving, but they feel the pressure has been taken off them now. They can see that other people who they consider to be brilliant at maths make mistakes and so, they too can make mistakes at its ok. I think you can't underestimate the value of being with other people of a similar ability.
Exeters model might be better in that they can do other a levels at the college next door. That may have done them in a bit on the league table side, but league tables don't tell a whole story.
The selection of a levels is very limited at KCL. It is really only suitable for those who want to study maths or engineering and not much else. My dc loves languages and would have done a language a level had one been on offer.
They do offer a good range of extracurricular activities (including a chance to do languages of their own choice for a term or two), They take them out on non maths related trips, Sport is compulsory one afternoon a week. Their epqs can be on any subject they like, they don't have to be maths based.
I think I'm leaning towards the idea that there should just be a few kcls - for the very gifted mathematicians. Those who actually really only want to do maths and associated maths subjects. And they should have a boarding school option as Exeter does, so that its not just for those who are fortunate enough to live within commuting distance (albeit very long commutes for some students) of the school.
I don't think they are actually a model that would work if there were too many of the KCL type school. You don't want lots and lots of very good mathematicians being siphoned off into such a focused curriculum - they are many areas of work that require excellent mathematicians, that require other subjects. But for the sort of student that my dc is, the school is almost perfect.
These schools won't in themselves cure the STEM crisis, but perhaps they will help spread good practise in teaching maths. KCL does do various outreach programs including a course for teachers who want to teach further maths. They also work with year 11s from other schools to extend them.
We are in a vicious circle at the moment where we don't have enough excellent maths teachers actually teaching, to inspire more children to love maths, so that they become maths teachers.