"Being School Governors, being close friends with the heads, getting involved with fundraising, daily harassment for a place." ?! I'm amazed at what chocolatedot has cited here. This is not how admissions for state schools work. If this is happening, it should be investigated. This would be in clear breach of a LA's admissions policy.
If there appears to be a bias towards "middle-class" parents going out of catchment to "better" schools - which I'm not necessarily denying - there could be all kinds of reasons for this. For one thing, middle-class parents are, broadly, more likely to be "bothered" and to make the active effort to indulge in "choice". They'll also know what to say at a school appeal, having done all the reading-up and, possibly, even hired a lawyer. And so on.
I'm not at all surprised that the private figures are higher in Edinburgh than the national average. I bet they are as well in Oxford, Cheltenham, Kensington and other places with a strong presence of well-heeled clientele. Doesn't really prove anything, except that where people are richer they can afford more luxuries. Which, let's be honest, it a bit of a "no shit, Sherlock" thing.