Wow, your child is me! I did exactly this, at 13, partly because of politics. And for years I stuck to my guns that it was the right decision.
But actually, with maturity - it wasn't. I was a very bright child and the private school was steering me towards probably going into the sciences, I loved them, although I was also good as languages, writing and loved art. I was often top of the class in several subjects (I know this as they tested us often). But I was bored, and although I had friends, I didn't feel I fitted in.
I had a great time at the state school, made friends and found the life and excitement I craved. Discovered weed, acid and partying, had a wonderful time. Academically I coasted. I passed my GCSEs on my intelligence and the education I'd received at the private school, not work. I became really unstuck at A-level and got terrible marks. I didn't get into any of the universities I'd applied for as a result and went to a local ex-poly through clearing, but I only lasted a semester, I was distracted by partying and had totally lost my work ethic.
I was so very bored at 13, I needed change, my life felt so controlled. The private school rules felt so stifling. But my home life was boring too, it never occurred to me that I could change that. I wonder now, if the adults and I had had more conversations to get to the real root of the problem, if we could have come up with different solutions to me leaving a school I was excelling at.
I've had a rich and varied life in terms of experiences and have some wonderful friends. But I've never achieved anything much career wise. Which is not what you would have expected had you known me aged 12.
I don't think it's at all true that bright children thrive anywhere. My son is at a school that reminds me a lot of the state school I was at. It's an OK school, not good or bad, but they're letting him coast. He's bright and they're not encouraging him or stimulating him enough. I'd send him private if I could afford it, against my socialist principles. I'm considering moving to a place with a better school but as I don't have much money it's a bit tricky.
Having said all that, what would have happened if I stayed? I don't know, it's easy to imagine it all would have been great but I don't know that really, do it?!