Your understanding is faulty, but the research that I think you're referring to was widely misreported, so that's not too surprising. What it showed was that if you match school leavers for A level grades then the ones from non-selective state schools do better by some measures than the privately educated ones. If I remember rightly, even that only applied to A level grades below the ones my current university requires anyway, by the way, i.e. it doesn't apply to the children who are most at risk of being unchallenged at school, so it's off the main topic. (I haven't checked just now, but if I remember rightly, once you got to BBB, you'd be overcompensating if you gave the pupil from the worst school an offer that was 1 grade in 1 A level lower than the offer you gave to the pupil from the best school, and at AAA the effect has gone altogether.)
But since you ask: suppose you take two young people who both achieve BCC at A level, and I tell you that the one from a comprehensive is more likely to finish with a good degree than the one from a private school. You want me to conclude that this means the one from a private school has somehow been invisibly damaged by the education, compared to the comprehensive school pupil. But what seems far more plausible to me is: those results were the best the private school pupil could achieve even with more support than the comprehensive school pupil received. Probably, the comprehensive school pupil is cleverer, or harder working, or less troubled, or in some other way doing better than the private school pupil who had more help but achieved no better grades. Now if we put them in the same environment, sure, we expect the one who achieved given results with less help to do better.
It's the wrong question. What parents want to know is: given this 4 year old, or 10 year old, will they do better at university or in life if I send them to this school or that school? Comparing students with the same A level grades tells you nothing that helps to answer that question. The results they showed are consistent with a quite different hypothesis: private schools, on average, get the same young children higher grades than those children would achieve at comprehensive schools. Some of that advantage may well be retained at university and throughout life. But, since not 100% of the advantage is retained after the pupil leaves the school (that's all we can tell), private school pupils don't quite keep up with comprehensive school pupils who have the same grades when they leave. Who's surprised? It would be absolutely astonishing if the only advantage private schools gave their pupils were via mechanisms that would last all their lives.