I'm a state school Oxbridge graduate.
There are lots of reasons why state school pupils are under-represented, particularly at certain colleges (my own, alumni including Michael Howard, Portillo et al was at best about 25% state when I was there, and most of those were grammars rather than the run of the mill comp).
Private schools are by and large exam factories. Certainly the good ones. My good friend went to St Pauls. Every single day of his education was geared towards exams, and getting straight As. He referred to it as an exam factory. They were also specifically tutored on entrance interviews etc. And the overwhelming presumption was every pupil would go to Oxbridge, or at a push a Russell Group uni.
Private schools from what I have seen, work on the basis that every child has Oxbridge potential, and work back from that iyswim.
Conversely, in 7 years of secondary school (state comp) I only had 1 teacher who actively encouraged me to apply to Cambridge. Most were of the view that I shouldnt aim too high My uni friends who went to comps had a very similar experience.
Have things changed in 20 years? Not from what I have seen. My son goes to the comp at the end of our road. In his speech at the start of the year, the head said he hoped a few of the new Year 7s (my sons year) might get to university. A few, from a year of 200! and not a few to Oxbridge...a few to ANY university.