Agreed, though not with everything. London Universities now have a massive problem in terms of recruiting good, but less well-off UK students from outside the M25. Such students are likely to prioritise places like Cambridge, Warwick or Bath. Imperial, Kings, LSE etc are doing some great work with London state school children but are otherwise caught up in a vicious circle. The very international culture of these Universities will not bother kids from either less affluent London state schools or academic private schools, who have experienced the same at school. However this internationalism (including how very hard some students work lots of anecdotes about students who sleep in the library) coupled with living in the Capital, might be off-putting for someone looking at campus or collegiate Universities in order to have a full University experience.
Where we might disagree is on the idea that Imperial will be a private school bubble. Exeter, say, might fit this description better. The DC we know aiming for Imperial, and we know a surprising number, are very bright and comfortable in an international urban environment. They will have schooled with the children of oligarchs and billionaires as well as children on bursaries and be able to take people as they find them. It is not unusual for London private school children who go to Oxbridge to find it a bit provincial, and we know some who have regretted not prioritising degree over University reputation.
The sub-set of students who participate fully in University activities will be smaller. However this group is be very diverse in a bunch of ways and very unlikely to be dominated by Hooray Henrys. I agree that as a result of both economics and because other Universities are chasing able mathematicians from northern state schools, this group will be under-represented, but there will be plenty of other forms of diversity.
(And not all foreign students are rich. When I lived in Asia, Imperial was preferred because it was far easier to find part time work in London. Ditto strong but less well off mathematicians from elsewhere in Europe, often prefer London's diversity.)