This is a very nuanced one, OP. NC to reply to you.
I am autistic. My IQ is 162, apparently. Not gifted but bright.
I had a terrible upbringing, very abusive. Went to one of the worst state schools in the country.
I got very high results, at GCSE and A-level. I was however relentlessly bullied at school as well as at home, so I was suicidal for all of that time. After I escaped my parents I barely attended school - in tje final stages of GCSE and for most of A-levels.
So yes, I got high marks. But was it a good environment? No. They didn't even notice that I was an abused child. It was the worst time of my life and more than two decades later my mental health still hasn't recovered.
I went to a very sought after University, got a very sought after graduate job and am now currently in the highest 1% of earners apparently. But life is still very hard for me now.
Would it have been better at a private school? There is no way to know. I expect the work wouldn't have been so boring and it wouldn't have been considered "elitist" to stretch intelligent pupils. But most of the people in my social circle now went to private schools and many of them were miserable and have long-term mental health problems as a result, too. Private schools are not a panacea: many are also rife with bullying.
So it depends what you mean by "success"? Getting good qualifications and a job? Mental wellbeing? From personal experience it appears school choice can go either way on both of those.
My primary concern for my children would be choosing a school that appropriately sets/ streams and really stretches a bright child so they aren't bored out of their mind. And even more importantly one that I was confident actually enforce an absolutely zero tolerance policy on bullying (many say they do but do not) so that any bully is removed to isolation at first offence and then if they continue, permanently excluded. As parents we should all be pushing for that above all else.
There is absolutely no excuse for it and I think far too often, still, "allowances" are made for bullies and the victims just suffer. It reminds me a lot of the debate in recent years about VAWAG and we really need to ensure all children have that same kind of protection where this is absolutely unacceptable. I think the ethos of a school - public or state - will be the most important thing to your child's wellbeing.
Despite my good results etc no I haven't achieved my potential, because these things have a huge impact that doesn't go away. I am not at all convinced that it would have been better at a private school though, I have to say. I have heard horror stories from those as well.