Education isn't just about what is in the class curricula, it is also about learning social skills, and just learning to deal with life. This is particularly true for a smart kid. I was an above average performer in school academically, but I had a tough time with social skills, and with learning to respect people I perceived to be my intellectual inferiors. Public school helped me with that a lot. I was exposed to people from all walks of life and learned that even if you are the Smartest Motherfucker in the Room(tm) that doesn't mean the other people are worthless or that you are better and oh, often you aren't the smartest even when you think you are.
Also it can be pretty hard to teach a really well rounded curriculum when it is just one person. You never know the stuff that may end up being valuable. For me? Various English class and speech class, bar none. They helped me overcome my fear of speaking in front of groups, and honed my communication skills. That is the second thing people will judge you on (after your looks) and it has helped me professionally plenty. All the math I did? Not useful, despite being in computers. Algebra is all I needed.
Plus for smart kids something that can be valuable to learn is that ya, maybe things move at a slower pace than you'd like, you need to STFU, deal with it, and do the work because actual jobs will have that too. You are not going to find some magic position where you are always stimulated, always challenged. Real life will be mired in BS too, so learn to deal with it.
I'm not trying to say public school is perfect, but in general I still feel it is by far the best option.
Also where I work (a university) we notice the same: The public school kids tend to do the best on entrance tests over all. A diploma and given SAT score gets you in the door, but you still have to pass university administered proficiency tests to determine what level of English and math you get in. You see a lot more home school kids getting stuck in the remedial classes than public school kids, including some smart home schooled kids. They just didn't learn what they needed to for university, in the university's estimation.