The socialization is the biggest concern for me. No question. If you can't relate and communicate well with others, you're going to have a lot of issues in life.
I don't know if I would want to add supplemental tutoring for someone in public school. Kids are already in school for 35 hours a week. Then you factor in sports/extracurricular activities, homework, time commuting, chores, and dinner. Doesn't leave a whole lot of time for anything else. One of the advantages of home schooling is everything becomes much more flexible. Spoke with a mom last evening that homeschools four children; she says they're usually done with school assignments by noon. So, you have a pretty big range to include social interactions and activities.
The biggest reason for homeschooling is you get to instill confidence, values, and character in your child. It's kind of a scary thought that the state, coupled with the influence of peers, could have a much bigger impact on your child than you could yourself. We don't want kids or teachers undermining principles that are taught in the home. Also, It's common for a parent to ask their child, "What'd you learn at school today." And they often respond, "nothing." I think it's sad when parents don't fully understand what their kids are going through - and they struggle to pry out any information about the happenings in a typical day. If we homeschool, we'd be directly involved with everything. You go at the pace needed - and you can customize everything. Not saying it's definitely the way to go. But, I don't think it's a completely terrible idea either.