I've studied in Cambridge myself, and feel like some context needs to be added here for American readers not to misunderstand something here: Admission to Cambridge & Oxford isn't related to family wealth in the same way as admission to "elite" universities in the U.S. For example: Tuition fees are in line with government-set rates and are no different for these institutions than other British universities (at least…
My observation studying in Oxford was that there was a big dispatity in opportunity, but it derived more from inequities in the quality of schooling available than unfairness in the admissions process. I met a lot of state educated students, but most of them came from well-regarded schools on upper-middle class areas that got hundreds of students into Oxbridge univeristies each year.
Very few came from schools like mime that couldn't even work out which entrance exams I needn't to take for the course I was applying to.