I think that I saw that piece of research, but interpeted it as being that if you could get the best A levels to get into the best universities without the assistance of cramming and support that private schools can offer, then you were going to do well (period).
Reading the discussion here, I think that the key is a school that you and your child are happy with. It can go wrong anywhere, at primary school it's all about the teacher in each year - if the teacher doesn't like your child, or your child doesn't like the teacher then you have a huge problem. Peers are also hugely important, and probably more so at secondary school. Bullying can happen anywhere, and so can drugs, bad language, and to be honest thuggery.
Of course no one wants to send their child to a school that they are unhappy with, and we would all do our best for our children - in fact I think that for many of us it's the concern about our kids going to school with kids whose parents really don't care that causes the most worry.
We should all have the choice of a good school without paying/moving house, but sadly that's not always the case.
Private schools may or may not offer what some people are looking for. For me the idea of hot housing my children into learning all sorts of things that I really don't care that much about isn't it. But feeling that they don't stand out like a sore thumb is. Being liked for themselves even more so. Feeling confident in their abilities also. I agree with Jimjams about the wrap around care issues, although this is available from some state schools too (not mine unfortunately!).
Oh, and I was taught by an ex-CERN researcher when I was at public school. Nice chap, obviously very intelligent, but had absolutely no idea how to handle 16 year olds. Result - I dropped out of physics A level. Teachers go to teacher training courses for a reason - teaching is something you have to learn how to do! Granted there are also fully trained teachers who are terrible, but I do think that some private schools (and some preps are notorious) take on unqualified teachers as a cheap option, and others (like my old school) take on PhD types because it looks good on the prospectus.
My mother taught at state school and sent me to private school. Then she came and taught at my school. I hated her for it!
I think that we should all send our children where they are happy however we can, but refrain from boasting to others who may not be so fortunate. After all it might start a heated debate