Homeschooled kids get plenty of social interaction with adults and other children in a respectful environment.
A lot of them do not.
Another nine year old has very little to teach another nine year old about effective socialization. School kids tend to be unnaturally cruel to one another as they learn socialization through trial and error rather than emulation of best practices.
The more important consideration is that adapting to the world at any age is largely a process of trial and error, whether it's professional, romantic, creative, etc. The world is indeed often unnaturally cruel as people adapt to it through trial and error, and removing those influences from the developmental process of a child does not in any way reduce their existence in the world, although it can set them up for a shock and even social disengagement later without repeated exposure to create coping strategies. Thus the position can easily be taken that not learning - and constantly practicing - these adaptive skills among your peers in youth is potentially disadvantageous as a long-term developmental strategy.
If someone decides to Home school their children, they need to take it very seriously and make sure their children get a proper education.
I think this is one of the two most important considerations. I've seen a few students come into middle school way behind the ball after five or six years of homeschooling. The parents wake up one day and say, you know, we're not really doing a good job with this. Susie, you're getting on the bus from now on. I don't think that's necessarily the right solution but it's probably better than continuing a poor homeschooling environment.
If you're going to home school, go all in. Don't half ass it. And if your kid ends up liking math, and you suck at it, you better think of something fast before they're asking you to teach them calculus when they're 14.
Cause one thing your homeschooling can't do nearly as well the school system is teach kids how to interact with mainstream society and thats one the most important assets you can have to be successful in this world.
This is the second important consideration. It's just a commonly understood truth in education that home-schoolers are more prone to undersocialization. Doesn't mean they all are, I know home-schooled kids who are very outgoing and social. But it's absolutely true, the potential is definitely there.
Some parents think peer socialization is underrated. I disagree wholeheartedly. There is a big difference between being a socialized introvert who is just inclined toward being shy, and being undersocialized and lacking the ability to read the social cues of your peers, body language, etc. Just my $.02.
If you're going to home school, try to bridge the gap and get your kid among peers as often as possible. Sports, gifted & talented, martial arts, clubs, camps, pickup games, neighborhood kick the can, whatever you can find. Kids absolutely need to be around their peers constantly, as well as kids younger and older than them, to develop a holistic personality and complex communication skills.
Guerilla mentioned self-directed learning, and that really is the most important thing, particularly these days, and it honestly has little to do comparatively with the choice of your child's education. You will always be their biggest role models and if you do it right they'll do it right too regardless of where they're educated.
Teach by example, communicate with your kids, be creative with them, paint and make crafts and snowmen and cook and show them how things work,
read to them every day until they read on their own every day, give them educational toys, bring them out into the world and expose them to stuff. Encourage their questions and answer every one they ask.