Just as some kids do wonderfully in traditional type settings, I'm sure some kids thrive in a HS setting.
Just a thought, but this sort makes it an either/or scenario, which is an odd way of looking at it. I'd think of it more like a Venn diagram - some are red, some are blue, some are purple!

I went to a traditional public school, but would have thrived in either environment (provided my mother wasn't my homeschool teacher - but that's not the point!)
We've never considered *not* homeschooling, but I've also considered the "what ifs" should something happen that would force Dd into the public school (we're talking both Dh and I would have to be deceased...but I guess it's a possibility).
In the end, I pretty much decided that bright children will figure it out and make it work - especially as they get older.
I was bored to tears in the PS I attended. But, by middle school, teachers had "caught on" and I was allowed to surreptitiously do my own thing, as long as it didn't interfere with the regular class work. So, while everyone else was doing spelling words, I was reading a book slipped under my desk.
In high school, I did my calculus homework in physics, physics homework in English, etc. I only paid attention in French class (a real challenge) and, on occassion, in English class - where I was often the only one participating in the discussion on The Scarlet Letter.
Adaptation and accomodation seem to be common tools of gifties - on either end of the school "debate."
Dd does get to "choose" some subjects, but some things are non-negotiable. She has chores, which are to be done before we start any classwork. And "sometimes" the kitchen really *is* closed - especially if we're on the third snack since lunch! It's how she learned to make her own pb&j.
As someone rightly said - homeschooling *is* a lifestyle as much as an educational choice. And, I would encourage you that if you do try it, give it a fair time period. It took me two years to really feel like we had a handle on things - and I had been planning to homeschool all along! Some folks give up after a couple of months, and consider it a "poor fit," but it really does take adjustment on ALL sides.
I've blogged some about homeschooling an only. Maybe our experiences might help a bit - I wrote the blog for my SIL who is considering homeschooling her dc, so it's more than a record of what we've done. You can find it here:
The Homeschool at Mingo's Corner HTH!