The difficulty is that you can't separate out the child's class, aspiration, and achievement level pre-entry, from outcome for the adult.
Example: I went to what has become "the state school college" of the traditional, ancient ones at Cambridge. It was over 80% maintained school intake. All well and good. But when you started to look more closely, the students tended to come from state schools in places like Richmond, so their parents had bought their way into a good catchment area; or from excellent - and extremely selective - grammars. And parental background was pretty solidly professional, too, in fact there was a joke that the access survey for applicants should read:
Are your parents: teachers/academics/other middle class professionals (delete as appropriate)
In essence, these parents were people who believed in state education on principle, but in every other respect were typical of many private school children's parents. I don't think there's any way to know for sure whether bright kids from supportive, educated backgrounds would do well at almost any school, and go on to earn well because that's what they've been brought up to expect and aspire to. It's hard to know if it's a private education that secures a place at a decent uni, which is where education really begins to have the potential to define life chances, or if the kids who do well privately would do well within a good state school, too.
I'm not saying the careful coaching, and extra help, and stretching to full potential in really good private schools isn't an advantage, I'm just saying that none of the evidence that might suggest that has much value, because even a family background that's pretty poor, on paper, might when examined more closely consist of a middle class single parent, so cultural capital is a definite factor. I think there are too many confounding factors for the info you're asking for to have any real value in evaluating outcome. The contribution is unquantifiable.