Anecdata
but I went to Cambridge (a few decades ago) from a state school. I was one of those kids that read every book they could get their hands on and my primary didn’t know what to do with me.
when I was at school there were free music lessons and I was put forward for all kinds of outreach and gifted and talented stuff.
i come from Lancashire.
when I got to Cambridge, I loved it. It was full of people who like me loved learning and loved reading and loved discussing things.
but even then it was apparent I was so, so behind just in terms of exposure to ideas and understanding the world. I’d never left the uk when I went to Cambridge, I didn’t really believe anyone actually lived in other countries and I’d never heard anyone speaking a foreign language who hadn’t learnt it in school.
my kids went to private school. I took them to French classes from age 2 (we did singing French nursery rhymes etc). They started Latin age 8, were offered Ancient Greek as an extra curricular age 9. My DS learnt three instruments and the prep school offered more ensembles than you could shake a stick at.
the difference between my state education (and mine was in the 80s with free music lessons etc) and their education is just night and day.
i don’t live near Cambridge anymore, but the Cambridge private schools are full of kids whose parents were state educated but went to Cambridge. They know the difference between state and private.