Welcome!
First, whatever decision you end up making, as long as your child knows that you are making decisions with his best interests in mind, for the health and happiness of his whole person, all other priorities will pale in comparison.
Second, no one knows him like you do, so after you've listened to and thought about what you and other people and information sources suggest, trust yourself, and ignore the judgementalism of other people.
Third, remember that no decision is truly set in stone. If it doesn't work out, try something different.
On to your specific questions.
We sent one through a tiny private for middle school, but it was a school with a lot of flexibility and multi-grade/age classrooms, with culture and values that aligned with ours, and an administration that welcomed our input on a wide variety of topics, and even spontaneously whole-grade accelerated our child (without telling us first--but that's a story for another day!), in addition to the single subject acceleration we requested. Two successive SSAs were teacher-initiated, the latter of which actually involved hiring someone to teach a single course to an extremely tiny class (just this side of tutoring).
Otherwise, it's been homeschooling for middle school. FWIW, the extreme extrovert is the one who was in b&m school, but was pulled out to homeschooling near the end of middle school, when even the very accommodating (K-8) school ran out of resources to meet DC's needs. That was not what DC wanted at the time, but in retrospect can appreciate was the correct decision on balance. So consider that there may be other ways to meet your DC's social needs, in the form of homeschool meetups, extracurriculars (possibly even through your local public school--in many states, that is an option), community service or faith-based organizations, etc. IOW, homeschool might not be off the table.
For us, I think the school climate/culture and attitudes of the admin and faculty toward children in general, and advanced learners specifically, were the most important factors. We actually left an enrollment deposit behind at an earlier school (which offered a fair amount of routine SSA and more dual enrollment opportunities than most schools of its size) after seeing some pretty negative teacher-student dynamics in action during a preview visit. Our thinking was that a certain amount of afterschooling supplementation in areas of particular interest could patch some of the lack in academic rigor, but no amount of persuasion or advocacy can truly change the educational philosophy of rigid leadership. To say nothing of the cost of actually damaging cultural practices. And if you're not paying (even low-cost) private tuition, those resources might be available for supplementing through high-quality afterschooling, at providers like Art of Problem Solving or Stanford online.
