Has every study also weeded out the effects of the huge advantages that most privately-educated children have anyway, regardless of their schooling?
Children who are privately-educated are often, if not usually, from families who are better off. I appreciate families make their own choices about where they spend their money, but to even have the chance of spending money on private school fees requires more money, regardless of how many 'sacrifices' are made - most families I know could give up every holiday they have ever had or are likely to have, and still not be able to afford private fees.
These children also tend to come from supportive families - parents who are willing to spend huge amounts of money on school fees, research all the different schools and possibly have their children prepared for entrance exams, are by their very nature the sort who value education and will support their children.
For example Xenia, I'm sure your children would have been hugely advantaged no matter where they went to school. They come from a well-off family, have a mother who clearly places great importance on education, live in a nice home with presumably enough space for them to study, and from what you've said in many posts they have had the chance to take part in many activities and hobbies and to travel widely.
So I would suggest that a large part of their life opportunities stems from their family background and not necessarily from their private schools. I'm sure they would have done well almost anywhere.