Well I can't speak for "people", but my DS pushed very, very hard to do his A Levels at one of these very selective private schools - one of those who are known in the press to get a significant number of pupils into Oxbridge, after doing his GCSEs at a state comprehensive - and after sitting the exams and getting through the interview he was accepted and did his A levels at this school (yes, huge financial stretch even just for two years, even though I consider us financially well off). This was very much driven by my DS, all we did was support what he really, really wanted. But as I don't think there are that many in our position, here are my observation on the differences:
It was a huge shock to the system going from a school where he was one of the high fliers (remember a comprehensive takes anybody) to somewhere where "everyone is really bright". Bigger shock than the facilities. He found it enormously motivating, although at times highly demoralising. On the other hand, amazing experience to have working lab equipment, and allowed to do dissection and really build up lab skills. At the stage of A Levels, it wasn't so much the breadth of clubs (although he did love having a politics club, and got heavily involved in it - one of the issues at the comprehensive was that debating politics if you didn't necessarily agree with the dominant narrative - eg Brexit, wasn't really possible, and no doubt would have done extra curricular drama, were it not for covid. He had always loved, but not had an outlet at the comprehensive since Year 8,- when the drama teacher left), but the ones that were specifically focussed on STEM subjects especially has he had decided his ultimate goal was medicine.
At the comprehensive there often wasn't enough equipment for the large classes, and the school was so paranoid about health and safety and the fear of what a bunch of rowdy teenagers could do with lab equipment that the absolute bare minimum required by the boards was done, and absolutely nothing more.
The chance to participate in clubs after school/lunch time where one could do dissection, research etc was mind blowing.
Also having a gym (and other amazing sports opportunities) rather than the small patch of concrete at his comprehensive where the boys played football - being the only place to let of steam. The comprehensive was/is scarily overcrowded and there is barely enough room for them to move, certainly not exercise.
But in terms of Oxbridge and academics
Going to the private school was what put him off applying for Oxbridge.
At his comprehensive there had been a lot of encouragement for the high fliers to consider Oxbridge, with trips to Cambridge in Year 9 etc etc. Not saying there was much actual coaching or that anybody had much of a clue, but it was sort of pushed. At the private school, after seeing what the competition was like, he decided he was not going to try. He felt there were far better candidates, and he was very relieved he didn't put himself forward for this when he saw who Oxbridge rejected. He also decided that he wanted a more hands on medicine course, with patient contact as early as possible, and that wasn't Oxbridge.
The one Oxbridge reject that particularly horrified him was the boy who, according to my son, was doing at least university level research and who didn't even get an interview for Oxford. My son got a 9 in Chemistry at GCSE (at the Comprehensive), and an A* at A level (at the private school), but he felt he felt the difference in both knowledge and passion between himself and this boy was vast. On the other hand, the boy had had the opportunity of these amazing labs (compared with the comprehensive), but even so, my son was sure that there was no contest. Now this boy was the most extreme case, but my son felt he was not the only one. That pretty much all the boys put forward for Oxbridge were better candidates than pretty much anybody at his comprehensive and he couldn't dream of competing. He felt it was very clear that in order to get into Oxbridge from this prestigious private school, you needed to be significantly more able than you did from the comprehensive, and that even then,many who would have walked into an interview and offer if they had been one of the very few from the comprehensive who applied, were knocked back pre or post intervew. He also felt that the comprehensive school had very much seemed to suggest that Oxbridge was the pinnacle of university experience, whereas at the private school, my son started considering questions like, did he really want shorter, more intense terms, and an essay a week? No question the private school knew Oxbridge much better than the comprehensive. But some of the coaching was also a winnowing - is this really right for you? And he decided it wasn't.
On the other hand, maybe you can say that with the advantage of the private school, you don't really need Oxbridge, at least in a STEM subject. The boy doing the amazing lab work will be doing world class research in a couple of years so long as he goes to any university with even half decent research capabilities, of which there are dozens in the UK. Whereas someone from the comprehensive may need Oxbridge to bring out their talent (close mentoring of the tutor system). In addition, despite my son's 9 in all his sciences and maths for GCSE, and having managed to do well enough in the entrance exam in these subjects, he felt he was distinctly behind those in the private school in both Biology and Chemistry (less so in maths, but he had an inspirational teacher in maths at the comprehensive), and it took him all of year 12 to catch up and start getting As in his A level assessments. although luckily he had two years to get the A*s he wanted. He started off Year 12 close to bottom of the class in both of those subjects, which was a bit galling, although he was determined he wouldn't and didn't finish there.
Another factor that really struck us seeing the make-up of the school is that the UK, and particularly London, is a magnet these days for global talent. And while it is a gross over-generalisation, the sort of people who can get a visa to come here from all parts the world based on their skills are often exceedingly bright, and often have exceedingly bright offspring. And a significant number of the children who get into these highly selective private schools are these offspring.
The days when the make-up of these schools were the English upper classes is long gone. That is, there was a sense when discussing the other DC in the school was that it didn't just take the very bright from the UK, they were increasingly taking a significant number of the very bright from the world, albeit the world who are making a shedload of money and have moved to London to do it. And if they have been legally resident here long enough to put their DC through school, good chance won't be classed as international in terms of university application.
Sorry, maybe overlong.