Somewhat ironic, to discuss my children's education, the funding of state education, private schools, NHS "underfunding" and PAs and then complain that the thread has lost its focus, don't you think?
The point about Cambridge, and the increased chances of privately educated students getting firsts, is that it occurred purely because Cambridge decided not to select on the basis of prior attainment, in an attempt to increase the proportion of a discrete subgroup of applicants - to "level the playing field" as you put it. When admission decisions were made on academic criteria, students from state schools and private schools had identical outcomes.
The problem with state education, as far as Oxbridge is concerned, is that pupils in the top quintile of nominally unselective state schools are 25x more likely to be accepted than children in the bottom quintile. I am not aware of comparable figures for medicine, but they are probably similar. This is clearly not related to funding, and does not have anything to do with private schools.
If WP applicants were "attaining excellent grades despite facing challenges along the way", medical schools would have been able to skew their intake in their favour without deprioritising academic attainment. But they were not reaching the high bar set by medical schools. So what has happened is that non-academic requirements replaced the academic. Medical school admission became substantially more complex, disadvantaging the pupils it was meant to help. UCAT is one example of this, it was intended to "level the playing field", instead it gives an advantage to applicants who have access to guidance, resources to practice, and if need be, to try again.
You may prefer to theorise about what makes a good doctor but prior attainment is the best predictor of postgraduate performance, without which career progression is not possible, and as McManus et al. concluded: "Passing examinations such as those discussed here is important for being a good doctor, and those who have difficulty in attaining such clinical knowledge will probably be less good doctors. Knowledge is generally preferable to ignorance, and clinical knowledge underpins clinical practice."