My only experience of private schooling was failing the 11+ when my mum tried to get me a scholarship into a local private grammar school when we lived in England. She maybe didn't realise that private tuition and coaching was something other parents were paying for to help their children pass the exam: she thought I'd get in because I was bright. I was completely lost as my primary schooling was no longer gearing pupils up for this.
I think there are state schools and there are state schools: there's sometimes an enormous variation in pupil demographics, resources and quality of teaching. This seems especially the case in larger cities. The postcode lottery is a very real phenomenon in Edinburgh. Some schools approach the comprehensive ideal with a good mix of pupil backgrounds; others are either populated with a preponderance of pupils from more prosperous homes; still others struggle with a large proportion of pupils suffering the effects of multiple deprivation.
It seems to me that when children are dealing with all kinds of problems in their lives through no fault of their own, then quality of teaching can only go so far. Unfortunately, household income and the educational attainment of parents are the biggest factors in determining how well kids do at school (leaving aside questions like additional support needs for the moment).
(By the way I'm quite enjoying this parallel meta discussion of education. All triggered by gembo's mention of John Swinney?)