Aspiration plays a huge part. And may be the greatest benefit, alongside a love of learning, DC received from a private education.
They went to a fairly new, quite "arty" prep school. Plenty of slebs on the school run! Subjects were set quite early so we had some idea where different children sat.
And guess what. Most of the DC whose parents were lawyers or other professionals have gone onto to good Universities and presumably professional careers, many of the children of (successful) arty folk, are doing arty things, often skipping University. Even though the stand-out clever children, in part helped by a bursary scheme, often had creative parents.
Aspiration means that DC got the aim high, work hard, from quite an early age. Not something that came from us, brought up with the constraint that you don't want to get above yourself, but from peers whose parents ran things and who expected to run things themselves. I assume that is one of the advantages that get from peers at Oxbridge.
I am also not sure what "clever" means. Some children are studious, which goes a long way to compensate for a lack of brilliance. Others are practical. My builder is not far off Piglet John in knowing the answer to anything to do with property, but achieved little at school. The trick is to find the right place. Oxbridge is not always the right place, even for very bright children. Young people need options and choices, not to be funnelled into somewhere that may not suit them.
I still think good sixth forms that deliberately welcome bright children from disadvantaged backgrounds are a better option. Not all will go to Oxbridge, event though the stats for the London examples are impressive. Better surely than an additional foundation year away from home for a relatively small number of children, which may or may not guarantee a place at Oxford.
There is enough deprivation in Oxford that a Oxford Sixth Form along the lines of the Kings Maths School www.kcl.ac.uk/nms/newsrecords/exceptional-results-for-kings-maths-school or the UCL supported Newham Collegiate www.thencs.co.uk/2018-a-level-results/ perhaps with support from local private schools like Harris Westminster, would work and improve life chances for a greater number. Other Universities could follow suit.