Dividing by school type is not particularly helpful.
The more academic private schools will see their role as preparing pupils for University study. OhDear has already outed my kids school on this thread, but safe to say it one that has been regularly mentioned and which sends coach loads to Oxbridge, and equal numbers to London. Their website suggests that "In 2025, our pupils took up places at all eight Ivy League schools, as well as Caltech, UC Berkeley, UChicago, Stanford, Bocconi University, and McGill University." In addition there were 72 Oxbridge places. The latter number used to be around 100. Some of the drop might be as a result of displacement as a result of Oxbridge's successful outreach programme, but some of the rest will be as a result of increasing competition from Universities elsewhere, mainly the US and London.
The school was pretty intense. DD had two members of the national debating team in one of her classes. You were expected to get the work done, but also be intellectually curious and challenge each other and the teacher, and then take part in a very active school life, including top level speakers invited by a pupil run society. Some of her peers were very bright indeed, but also had the advantage of being able to hit the ground running when they arrived on campus.
A friend had two girls, the second of whom did not manage to get into one of London's more selective private schools. Right at the start the younger girl's school briefed parents on homework and expectations. Spoonfeeding if you like. The syllabus was taught and the work set and if the girls were diligent they would get the grades. The supposedly "better" school was less organised. The pupils selected were very capable of getting the grades, so there was time to focus on stimulating imagination and curiosity.
There is a lot of teaching to the test in both sectors, rather than having an aim of using sixth form to prepare the most able pupils for University. The most impressive students will be those who, because of a poor teaching environment, have had to a large extent to teach themselves, so arrive at University ready to do the same. Second though might be pupils from privileged educational backgrounds who are used to using both teachers and fellow pupils as resources to help them pursue intellectual curiosity.