I am, I suppose, quite clever, although far from a genius: I was, indeed, top of my class in (state) secondary school, and, yes I went to Oxford, and, yes I have various degrees including a doctorate. And all the rest of the academic stuff, books, research papers, etc.
(I would be interested, OP, to see what 'research' your son is pursuing in quantum field theory. Perhaps you are over-egging this particular pudding just a little? Reading Wikis or Quora is not really research.)
My children are clever: all PhDs, some academic jobs, some not; successful in many ways, but, well, phew, no genius there.
And now I have some really clever grandchildren. Is a six-year-old who reads Einstein as well as Rowling and Tolkein (and Shakespeare) a genius? (Others, well, reading and writing in several languages aged eight, so on.)
No, far too early to say.
I did not go to private school. I did not send my children to private schools. I do not advise that my grandchildren go to private schools.
British state education is far from perfect: classes too big, too much rote learning, no real foreign-language teaching, unqualified maths teachers (!), SATs (urgh!), OFSTED (aargh!), and so on. But British private education is much worse, despite smaller classes etc., etc. Trust me.
In the normal run of life, of course I have been close to many products of (the more expensive end of ) UK private education. Close up, as it were, these people, however clever they may be, are for the most part hugely unimpressive, intellectually, emotionally, psychologically. This is true whatever their political leanings or general ethical views. Moreover, a significant proportion of them, when push comes to shove, are aware of their failings ... and mostly, again in my experience, these aware people blame their education. (Of course there are also the 'Never did me any harm!' people too. Self-awareness is too often too hard-gained here as elsewhere.)
So, in short, my advice is, send your clever child to the best state school in your neighbourhood. Encourage him or her to read books, write poetry and stories, talk, play, make friends, code, visit museums and galleries, make music, engage in sport ... all the usual for the mens sana in corpore sano that private schools so spectacularly fail to accomplish in their alumni.
Your child, if he is as clever as you think, will thank you for your choice. Such has been my experience, anyway.