If you want to make school (either public or private) work, you need to be very clear in your own mind about what it is you want your son to get out of it. And keep in mind that what parents want and what students ultimately decide they want can be two very different (and sometimes incompatibly different) things.
If you're wanting an excellent academic fit, you may be able to get it through some sort of combination of acceleration, gifted services, and teachers who are able to offer appropriate differentiation though it's unlikely. But I am going to be honest--how well these things work can vary from year to year and even within the same year. They are heavily personality dependent--that is, good solutions for a particular student seem to come about when individuals are motivated to be flexible and creative and not because of programs or policies.
If you decide to homeschool, you are in control of academics. This means that you can go as fast, as deep, and as difficult as you want. But I will warn you, with kids as bright as your son, homeschooling at their instructional level can be a wild ride and it will leave them extremely out of sync with age peers if they ever decide they want to go to school for social reasons. That said, I did this with my son in from age 5-10 and again from age 12-14 (see below) and would do it again in a heartbeat.
If you're wanting a good social experience, you will need to consider where your son would fit into the school community best. Is it with intellectual peers? Is it with age peers? Somewhere in between? Unfortunately, even if you decide that intellectual peers would be best, there may be none, or very few, at your son's school, and accelerating an HG+ student into a classroom of average students may prove to be a mismatch. Regardless, if you decide to enroll him in school for the social piece, you will need to let the idea of perfect academics go or else you will go crazy. Ask me how I know.
Our son, who is now 18, was homeschooled K-4, skipped 5th, did 6th at a tiny private school, skipped 7th, and did 8th at the same tiny private school. The skips did not provide increased intellectual challenge, though they did provide EF challenges. It's hard to say how the skips affected his social experience--he didn't really fit in at the school for reasons that I see now weren't related to his age.
Anyway, he homeschooled for the next two years and then decided that he wanted to be "normal," so he enrolled at the local public high school in 9th grade with age peers. We were both very clear on the idea that he was doing this for purely social reasons, and anything good that came of it academically would be icing on the cake. Halfway through 10th, he had had enough, and so we withdrew him from the most "meaningless" of his classes to homeschool. We are lucky in this state to be able to use the public schools part time, and this is what we have been doing ever since. He gets his fix of hyperintellectual academics at home, and he gets to be with his friends every day too. At least he did up until last week when school was cancelled
