My ds has applied and interviewed for Oxford. We find out in January if he gets a place.
He attends a state college that performs well below average and almost never sends anyone to Oxbridge. They post their highest results in the local paper on results day: last year one girl got an A-star in geography and that was the only A star in college.
I don't know if state educated Oxbridge dc are more intelligent than private, but my ds certainly has needed much more self discipline, self motivation, and resilience than a dc who is catered for at private school.
As an example, ds takes further maths but at the end of year 12 his teacher told him that in year 13 the college wouldn't have the staff to teach the usual three lessons a week. Ds only gets two lessons a week in further maths and self teaches the additional work.
To make this possible, ds spent the 6 week summer holidays before year 13 teaching himself all of year 13 maths so that this year he could focus more time on the further maths.
There were no Oxbridge practice interviews at college, no personal statement help. Ds only has two friends at college that are going on to university. The others were kicked off their a level course after year 12, doing apprenticeships, etc.
So I do think it takes different skills and character for ds. He walks a fine line socially between being ambitious but not obnoxious and alienating his friends who won't come near his grades.