In addition, the colleges have a lot of data about the people they interview and how well they do during the degree programme.

My understanding (based on a discussion with one Natural Sciences admissions tutor at one Cambridge college nearly 20 years ago, so strictly speaking this may not be true in general, but I'd be surprised if it wasn't common) is that during the admissions process, including interviews, applicants are scored so they can be stack-ranked, and the top N given offers. Then, for the students that are accepted, and get the required exam results, the college also records their marks at each stage of their degree. To verify the admissions process is fair, these marks are compared with the original interview ranking, expecting that interview performance is (on average) correlated with later degree performance.

I don't know if they go further and build models to suggest the correct offer to give different students based on interview performance, educational background, and other factors, but it seems at least plausible that one could try that kind of thing, and have the data to prove that it was working.

Anyway my guess is that of the population of people who would do well if they got in, but don't, the majority are those whose background makes them believe it's "not for the likes of me", and so never apply, rather than people who went to private schools, applied, and didn't get a place.

(also a Cambridge alumni from a state school, FWIW),

All these Cambridge alumni with this dodgy Latin, 'smh'! You're an alumnus, or identifying as an alumna! (Identifying as many alumni at a stretch, but then still not 'a Cambridge alumni'.)

(alumnus not of Cambridge, but from a state school, fwiw)

('people called Johns, they go the Cambridge?!')