It is true that children scoring the same CAT scores on entry to independent schools achieve somewhat higher grades than children with the same scores in state comprehensives. The MidYis predictions take the school type into account for example so as to give more accurate predictions. So there is definitely a higher 'value add' for independent schools, but I don't think it's anywhere near as pronounced as 'average' students getting top grades, to be honest. (I used to know what it was in terms of average increase to GCSE grades, but I've forgotten).
However unpalatable it is, the average intelligence of independent school kids is likely to be higher than those at state comprehensives (even true comprehensives) and this will also be a factor in their over-representation in top universities. Some of the reasons for this discrepency are:
firstly because the scholarships/bursaries do succeed in attracting the very brightest children who would otherwise not have been able to afford it;
secondly because some parents with a very bright child make extra sacrifices to send that child (and often not others in the same family) to private school because they believe, rightly or wrongly, that a selective school is the best environment for them.
thirdly because the average income of the parents is higher and for some reason this correlates quite strongly to educational outcome in all types of school - so for some reason (be it nature or nurture) children of rich parents achieve better, quite likely due to higher average IQ - caused by either genetics or nurture (more enriching upbringing), who knows) or maybe by the extra help being available when needed. This doesn't have to be a huge effect and is the most controversial I think, but it does seem likely that it is there at least in some measure.
lastly because many independent schools (more than half I think) are selective to some extent and filter out the less able applicants this will also contribute to the average intelligence of the overall independent school child being above the national average.
Having said all that there is clearly a huge overrepresentation of private children at top universities that is unlikely to be anything like fully accounted for by the above (quite possibly even including the extra value-add) and is probably more down to school and parental expectations and guidance.
That said, I do think having been taught well and achieved high grades and gone to a top university that a child might not otherwise have got too isn't necessarily a disadvantage. Lots of children in that position still go on to do well - having learned how to learn they are able to continue to surpass themselves. As I think all of us tend not to fullfill our potential there's plenty of scope for people to cope with courses that maybe they wouldn't have been able to access if they'd got lower grades from another school. If they've had a good well-rounded education many people can build on that and take it further.
At university (Oxbridge) the people I knew that struggled most were evenly spread across school types. The privately educated struggled mainly not due to lack of intelligence but excessive partying; those from state comprehensives that I knew well struggled academically most (but I also knew several who didn't at all); and those I knew from state grammars also struggled academically and also quite a lot from the change to independent learning as well. All anecdotal of course, and I'd really hesitate to jump on anyone's anecdotal observations (where people tend to see what they want to see in any case) to support a theory, but it shows that not everyone observes privately educated children floundering through innate lack of intelligence.