It was my understanding that the issue is that by the very nature of the beast, privately educated DCs are far more likely to have been given individual attention, tightly focused teaching tailored to that DC's weaknesses, all in far better equipped classrooms in much smaller groups than their comprehensively schooled counterparts.
Surely you'd be demanding your mine back if your DC wasn't reaching their potential in that environment!
I have a friend with DC in 'good' private schooling. She had to admit that when the head of the prep called her in to tell her her Y6 DS was barely making the grade for the secondary school tranfer, in both Maths and English, she realised that as the DS in question was working to his utmost ability (as there was no other option at the prep!)- there was no more to give. What you see it what there is. Whereas in the majority of state schools, you cannot be certain that the DC hasn't got more to give, it's just that the resources to extract that 'more' might not be financially available.
It's the flip side of the argument that a A private student might have been capable of A ++ had it been available.
Finally, if I had a DC in a private secondary, I'd not risk the move to a state 6th form. Surely that's when it pays dividends. As another thread points out, re Eton entry, to get in you have to be on a completely different educational trajectory to the rest of us. You have to pick which 'train' to be on and stick to it til it reaches its destination. The risk of discovering that your DC only got a bunch of A* because they had a teacher standing 4" off their shoulder for 3 years at GCSE, when they go on to perform really badly at 6th form because they're expected to carry on with their independent learning as per state comprehensive would be bitter indeed.
I'd stick with the private and believe that a government made up of rank after rank of ex-public school boys, all depending on the right-wing vote isn't about to scupper the golden futures of their voters privately educated DCs.