Homeschooling your kids can be a great thing especially for a family with a pilot. Although there are some downsides. Think about it, DH is out for god knows how many days, mom is with the kids 24/7 non-stop. DH comes home and wants to relax sorry we have all the kids at home. To be a good homeschooling parent you must do tons of research, curriculum, child learning styles (if you have more than one kids you will probably have more that one learning style) and state laws just to name a few. HSLDA.org (website for Homeschooling laws)
We homeschooled up until this year because my dh is a first year FO. So you know he is getting paid peanuts, so I have to work to make ends meet, plus we have 2 kids that need services which insurance at the new job won't cover but they are offered at the school.
Socialization if homeschooling is done properly is not going to be an issue. There are homeschooling groups, co-ops, sports, etc... My kids often complained that they had too much to do and just wanted to hang at home and play video games.
Oh, anyone that wants to say the American school systems are good is full of crap. One thing we do is host exchange students from around the world. All these kids come here in shock because they don't even do a fraction of the work that they did in their home countries. Example my exchange student from Germany who is with us right now is taking senior classes and blowing his classmates away. He is taking AP Calculus, AP English, etc... Oh, and he is supposed to be taking sophomore classes.
If you want your kids to have the best education, go fly international and put them in schools in Europe, South Korea, or Japan. We have hosted students from all of theses countries and they are brilliant kids. I am hoping dh gets to go international some day for this very reason but my kids will probably graduate by the time this happens.