DD3 is an only so far (and likely permanently).  For us, homeschooling offers me benefits--I enjoy being with DD, talking & thinking about a wide variety of things, and I can only imagine it getting better & better when she is into more & more difficult topics.  She is a challenging and sensitive child and I get her; I would hate to send her off to a teacher who might not get her.  I want to stay connected to her and know what is going on in her life.  For her, I want the incredible flexibility of education that homeschooling affords, which is totally lacking in most schools.  I want her to learn new and interesting things, rather than be drilled in things she has known for years.  We are in a relaxed homeschooling coop now, so DD does get to experience the class-with-peers thing, and we will likely continue that or even dual enroll with public school for non-academic topics (gym, art, music) when she is technically school-aged.  
I don't see many benefits to most public or private schools for a HG+ kid other than as a babysitter.  I guess this is mostly a reaction to my own experience being PG in public school.  But I stick by what I said:  If you aren't into it, homeschooling isn't the answer.  An unhappy/overtired/disgruntled/frustrated/resentful mother will do more damage than good.  Homeschooling really isn't for everyone.  Oh, I guess I would say that public school (if not private school) can have the benefit of exposing a child to a wide variety of people, which is something that homeschooled kids do not always get.