We're doing the opposite next year, transitioning from a public gifted program to a private school that has a reputation for being challenging.
The single most positive thing about public school for us is that the kids have true peers, geeky and gifted kids who come up with awesome crazy ideas together. Even in a GT program though, there are only about 5-10 kids per grade level that are truly GT. They manage to find each other.
The public system is cumbersome. The quality of education depends entirely on the teacher from year to year. Our school is unwilling to accelerate. One of my DS got a 98 on the end of year exam they gave for math in September and the school still refused to accelerate him, even after my various pleas, because he didn't get 100. They have been willing to give him some pullout support (but he still has to do the basic curriculum work) and he does have two little friends in his class who are equally interested in math. So they're making an attempt, just not going as far as I would like.
We are trying private, only because we are hoping that the individual attention will make a difference. I am concerned that lack of true peers may be a problem, but am hopeful that there will be at least as many as in their public school, and I'm confident that they will find their tribe. Your son is still quite young. Mine were in about 3rd grade when they started to make their own firm friends, beyond just the sons and daughters of our grown up friends.
There are no easy answers and there is no perfect school. It sounds like you are on top of it, and know your son well. This is what will make the difference for him in the long run, rather than any specific choice you make.
