Thedolly... OK deep breath, fortunately (or maybe unfortunately?!) I cut and pasted it all and it is still on my pasteboard.
If it's any use at all...
I was both privately and state educated. When I was state educated, I went to a school in a very nice semi rural area, an area where the schools today all carry good / outstanding Ofsted results. Again a three tier system, (which I personally think is excellent and very beneficial to pupils). I work in a private school, and my dcs are being state educated. My social circle is a pretty eclectic mix of private and state educated friends. I therefore feel that I am reasonably qualified to offer you my thoughts, and I hope they are of some help.
You will have to forgive me for any sweeping generalisations here BUT this is purely based on my OWN observations and I have found that:
Many many private school pupils (and ex-pupils) come across as amazingly confident. They basically tend to hold their own in conversation really well. They're very positive and seem to command an appearance of success, IYKWIM. State pupils seem to come across as less confident and less socially 'in your face'. (It could be argued that this is a good thing. ). However, I don't think this means an awful lot. There was a hugely positive vibe among my friends at state school, and we all had strong aspirations and wanted to do well. Especially at sixth form. In my lifetime I have met many many strong and interesting people who have come from state school, and despite our school being in a nice area there was a fair degree of diversification among pupils. I really didn't and still don't see this as a bad thing. Many private pupils seem to eventually emerge from a very cloistered world and I noticed at uni they would all stick together, (and tbh seemed repelled at the thought of mixing with kids from state school! .) even now I feel that there is a social divide and in particular one social group I know really sticks together in a bit of a 'them and us' way and rarely fully welcomes 'outsiders'!
Looking at it all now that we've all grown up (!) I have often thought there is NO WAY in which you could tell which 'section' is the more successful. The universities chosen by my social peers were the same ones whether my friends were private or state. In fact, now if I had to say one way or another which group has done better overall, now we're all in our thirties and forties, I would say it is my state educated friends. If we're looking at aspiration and success in a fiscal sense then I can honestly say that many many of my state ed friends are doing VERY well, thank you. I won't bore you with further details of their careers here but I'd happily list a few examples if you wish.
You could argue that nearly all my state-ed friends are MC and most of them had fairly affluent parents. But I believe my words hold weight because you are facing a similar situation with your dcs.
What this all boils down to, IMO, is that the aspirations of the individual are key here. Parental background is a massive influence too; Personality, aspiration, confidence etc are hugely affected by parental influences. As far as the school is concerned, well yes there is no denying that private schools seem to nurture a huge proportion of healthy happy and well rounded pupils, and I suppose the 'ideal' situation for many would be to feel that one's dcs are benefiting from the excellent standards of individual attention and pastoral care that many private schools offer. However, if private schooling is not possible, there is another way, and it isn't necessarily headed to failure. WHATSOEVER!
P.S. I'm sorry if I'm being bias towards state here; we all know that private schools usually produce excellent results and successful individuals. But it's clear from your OP that not everyone seems to realise that some, many, state schools have and continue to produce excellent results and highly succesfull, well rounded individuals too!
Rambling over, well done if you've got this far.