Fourteen or fifteen years ago when my children were choosing a university and college, there was no formal careers advice at their (major) public school but even then, staff (from house matrons upwards) knew from experience to guide pupils away from applying to Cambridge. Conversely, the comprehensive school near their home, which was ranked in the top 50 for Oxbridge entry, almost invariably directed their pupils to Cambridge, and not Oxford.
This was around the time that Parks produced Academic Performance of Undergraduate Students at Cambridge by School/College Background. This short paper demonstrated that there was no significant difference between the academic performance of students at Cambridge from different types of school. But Cambridge was already setting out to increase the proportion of state educated students. Parks' successor, whose name I don't recall, had a slide presentation intended for admissions tutors setting an AS-level UMS mark cut-off two or three points higher for private school applicants.
In due course, Cambridge published an Access and Participation Plan which contained two quite distinct aims: One to increase the proportion of students from disadvantaged backgrounds, and another to increase the proportion of state-educated students. The only justification for the second was that "... each of the under-represented groups identified within this Plan appear in far greater numbers in state maintained schools, as do students from low income households who are not identified by any of the measures currently available to us." In other words, if we vastly increase the proportion of state educated students we might pick up a few applicants from disadvantaged backgrounds that we might otherwise miss. To put this into context, when the plan was conceived 40% of POLAR4 Q1 admissions were from comprehensive schools, but 15% were from independent schools.
Raising the bar specifically for applicants from independent schools had a predictable effect. As the proportion of state educated students increased, their performance relative to privately educated students decreased. In 2017 the university stopped publishing examination data for students according to the type of school that they attended, but the schooling gap continued to increase.
After this behaviour, it should be astonishing if the university were to receive any applications from independent schools. I think it does because Oxford belatedly, and to a lesser degree, has gone down the same route. But they both lose out to US universities, and to a number of UK universities that are snapping at their heels, deservedly.