Ipad,
I would also say - and this is from someone who works in, sends her children to, and believes wholeheartedly in, the state system - that the gap between state and private options varies hugely across the country, or even within a single town, and it is often this gap which is important when making choices, not the absolute quality of the private school.
So in some areas, using the private system does not give any more likelihood of 'ultimate success' than a good state option does in another area, and may even give less. However, it does give a very much better chance than the stae option available to that family at that time, and iut is on that school by school (not sector by sector) comparison that choices are made.
The OP has stated that she would cheerfully send her child to a state school elsewhere in her area, but that living where she does, her best chance of an education comparable to that available in the good state schools is via the private system, and I understand that, though I would still urge her to take a really good, close look at the state option and examine the education available there to the subset of children who are like her own child, rather then being swayed by rumour, overall results, and the type of cohort.
I would be posting differently if she was saying 'there's a brilliant state school just opposite my house but I want to send my child private because I think that might give her a bit of an edge'...and even then it would depend if she was talking about St Somebody's tinpot private with nice braided uniforms or Winchester.
Where I live, going private for primary buys you nothing over and above the local state primary (except outmoded teaching, more traditional uniforms and - critically for most of the people who send their children there - a degree of social esegregation and 7 years of intensive coaching for the local 11+ superselective system). However, live a mile across town and the calculation really is very differnt, simply because of a wide variation in primary school quality.