ChipStix State school educate children who make it to university get higher degree classifications than their private peers.
No, they don't. You're probably alluding to the publication by HEFCE with the embarrassing typo (see here for example).
Some research finds that if you compare state and independent school pupils with the same A level grades, the state pupils do better at university. The difference is larger at low grades (below the levels achieved by most independent school pupils) and becomes tiny to non-existent, depending which study you read, at grades BBB and above. Cambridge's own investigation found no such effect in its admitted students, almost all of whom have AAA or better, for example. There's reason to believe that, if anything, today's contextual offers over-compensate, on average, though they're a very blunt instrument.
Suppose for a moment, though, that this effect did exist even at the top end. It's still not what you want to know, if you want to compare the education offered by schools (for example, if as a parent who has the choice, you want to choose a school for your child). What you want to know is what degree results would have been obtained by the same pupil if they'd gone to each school, or realistically, by groups of pupils who were comparable when they entered the respective schools. Let's do a thought experiment. Suppose we have matched groups of pupils about to enter state school A and independent school B: pupil a1 is comparable in every way we can think of to b1, etc. but a1 will go to A and b1 to B. Let's order them roughly by ability and parental support and whatever else you think matters for degree success apart from school - let's say we expect a1 to do better than a2 who does better than ... down to a50, and similarly for the bs.
So, let our students go through their school careers at schools A and B respectively, and pause when they've got their A level results. How did they do? Well, if school A is offering a worse education than school B, we expect the as to do a bit worse than the bs. We still don't expect them all to do the same; in the thought experiment, we expect the as to be roughly in the order we had them in before, and similarly the bs, but e.g. maybe a17 might get ABB while b17 gets AAB, who knows. Probably a1 and b1 both get AAAA though (oh, add some stars); there's a ceiling effect, anyway - among the best students, even if their education in school B has been better than in school A, we don't expect that to show up in A level results because there's nowhere to go. ("Clever children will do well anywhere" dontchaknow.)
Now pick some set of grades, let's say BBB, and pick out all the state school pupils and all the independent school pupils in your originally comparable groups who got BBB. Because the independent school group did better overall, these students will be higher up the list of as than up the list of bs. Maybe the ones who got BBB and went to school A are a30...a40, while the ones who got BBB and went to school B are b38...b48.
Now they all go off to university. (Just to avoid making this too complicated, they all went to the same university to read the same subject :-) Pause when they graduate. Look at our BBB students. Look, the ones from state school have done better than the ones from independent school! Seven of the students from the a30...a40 group got 2is or above, while only six of the students from the b38...b48 group did. State education is better!
Ah. But what happened to b30...b37? That's right, they got better A level results than BBB and a higher proportion got 2is or above. Meanwhile a41...a48 got worse A level results than BBB and a lower proportion got 2is and above.
So now which school did better? If you compare the students who came out with the same A level grades, school A did, but if you compare the students who were comparable when they went in, school B did. As a parent choosing a school for your child, the latter is what you care about, because you give birth to a child, not a set of A level results.
You will see this "state students with the same A level grades do better than independent students with those grades" effect if anything less than 100% of the advantage that independent schools confer on A level grades is maintained through to degree results. It's not that interesting to anyone except university admissions officers (and they know all about it, believe me).