I have now read the stats and I still don’t understand the argument.
Setting aside school type for a moment, female undergraduates got fewer firsts and so did certain groups of BAME student, if I have read these stats correctly. I would like to hope very few people would even try to suggest this is because those candidates are less bright.
When I was a student (more than 3 decades ago) there were relatively few women getting firsts in almost all (possibly even all) tripos subjects at Cambridge. It was suggested back then that this is because a certain level of self confidence is required to exhibit the kind of standout bravura performance needed to get a first, rather than the kind of steady, conventional and safe approach that tends to get a 2:1. Most tripos subjects have also traditionally been dominated by exams, with much riding on one week of old-style finals, which it has been suggested can tend to suit typically male working styles more.
We know from research on school assessment that changes to assessment styles swing percentages in relation to male and female achievement.
There are clearly far more factors in the mix in relation to attainment at the university than raw talent. I’m unconvinced that drawing a straight line between achievement at university and ability either in relation to students from certain school types, gender or other diversity characteristics is possible.
Any attainment gap opening based on school type could well be because of a mixture of prior educational input, lack of confidence/struggling with the unfamiliar environment. What these stats are perhaps really suggesting is just that more efforts are needed to support students from lower socioeconomic backgrounds and enable them all to attain at the level of their potential.
I sent my own dc to private school and I and other private school parents have to accept that in having done so we have purchased privileges for them. They have smaller classes and more individual attention. It is to my mind entirely fair for those privileges to be taken into account when it comes to Oxbridge admissions.