I homeschool an introvert PG only DS7. I came to homeschooling by default. Public schools are not option in my state. DS had already been in two private gifted schools for part of pre-k and kindergarten. They could only accommodate him so much with his rapid acceleration so I knew eventually we would probably end up homeschooling him at some point but I didn't think it would be so soon.
Homeschooling is a viable alternative and may be the best option in some cases. It may be the lesser of two evils or the least-worst case scenario. Another point to bear in mind is that the situation may change over time. Homeschooling is not necessarily for life or solely with you directing the studies. What works one year may not the next. Further your approach or way to homeschooling often changes with time. At the end of the day, you could always treat the homeschooling as a trial and see how it works.
My concern with socialization was on my mind when we decided to homeschool, but really it was the situation with the public/private schools that left us with little choice. I then decided that I've got to try to make the most of the situation and see what happens with it. I've found the socialization to be a less concern than others (i.e. rapid acceleration or math material). And I've found non-gifted parents who homeschool to say the same thing - that socialization is less of concern than others.
I think the other aspect to bear in mind with these kids is that they don't always have to be with their chronological aged peers. They can become friends with someone younger or older. Or even an adult.
I take DS to an open gym at the Y for homeschoolers once week and Y to swim. We try to meet other homeschoolers as much as possible. We visit the libraries every week; we try to arrange some playdates or go to the park/playground. Parks & Rec depts sometimes have things going on. There's usually things around if you look.
I think you'd find both gifted and non-gifted parents will say that the opportunities for socialization are far more open and varied with homeschooling than children have in any formal classroom setting.
