C.C.
Excellent answers below. I think a common misconception about homeschooling is that it's exactly like public school, only it's at home. That's not the case at all, actually. Once kids can read well and have solid reading comprehension skills, most homeschool books are directed at the child, not at the parent. My 6th grader rarely needs help from me, surprisingly enough. Typically, she may want to confirm with me that her understanding of a math concept is correct, and we will review it together. But in terms of history, literature, social studies, and art, most of our curriculum is very well explained and self-contained. She turns in her work to me, and we look over it together. Just like any teacher, I'll point out areas where she could have improved (and generally, she will take her paper back to her desk and do exactly that). We will discuss ideas for further study, and she will spend her afternoons (after her "regular" school work is done) doing a "deep dive" into subjects that interest her. She has become adept at searching out primary sources online, and finding the right kinds of books at the library.
In any case, we know many families who are homeschooling older children (junior high and high school aged), and it's much the same for them. One of my employees is a high school Senior this year, and has been homeschooled since preschool. She had a 2200+ score on her SAT test, and has been accepted at every college she has applied to. Neither of her parents has a college degree. (For the record, her older sister had similar scores and attended her first-choice college as well.) They do belong to a homeschool co-op, but it's mostly for socializing and doing crafts. :)
Bottom line, homeschooling is a completely different animal than public (or even private) school. The curriculum is different in that it is much more complete, and doesn't really require a teacher to interpret it.
That being said, I went to a very good girls' boarding school (just the next town over from where you live ;), and I'd love to send my girls there. But for $50K per year per girl? Yeah, I'm pretty sure I can provide a more than decent high school education for my girls at a MUCH lower cost. Think of the college classes they could take for a fraction of that cost! Would it be as wonderful an experience? Probably not. But I have to be realistic and try to determine how to get the best bang for our buck. Unfortunately, the public schools where we live aren't great, so we have to be creative.