I think the socialization argument is the hardest one to dislodge, and the easiest one for doubters to cling to.
My aunt is convinced that dd1 is really missing out since she's not in a school. I think for people of that generation (over 50 especially), they don't realize how different school is, especially in the early elementary years. They don't know that Pre-K is what they did in K, and that K is hard-core first grade. They don't realize that recesses, PE, wiggle time, naps etc. have been decreased or phased out. Heck, I've a friend whose second grader didn't have recess until the school day was OVER. How can a kid socialize and wiggle with that sort of set-up? We have public-schooling friends who are struggling with their public school's policy of _not talking at lunchtime, at all_. So, when is all this 'socialization' that kids are supposedly getting at school? Maybe half an hour or 45 minutes of recess in the entire day, the rest of the time to be following teacher directions? For a day which starts at 8am and isn't over 'til 2:30 or 3:30 (not counting bus time)?
So the first step is to remind them that school isn't what it used to be.
Second, let them know what your child is doing for socialization. For us, when dd1 was in K, that was:
Swim lessons 4 days/week (1 hour/day, mixed ages)
Art class (grade-specific, 1 hour/week)
Play date, Wednesday - 3 hours with some other homeschool kids
Library/play date, Thursdays - 3 hours with a group of friends
Church class, 1 hour/weekly
Monthly stuff: Cloverbuds (4-H, 1 1/2 hours once a month - activities and learning stuff, kids 5-8)
Seasonal stuff: Gardening classes at the local nursery (weekly, 2 hours/week, mixed ages of kids), lasts 2 months
We don't even do that much, but we've a good mix of structured and unstructured socialization time. This doesn't include random friend playdates (which typically happen a couple times/week on top of the above), nor does it include social time with her cousin, nor does it include all the visiting she does with Grandparents, elderly friends, random acquaintances that she knows well by now (clerks at stores etc.).
So if a public-school kid gets 4 1/2 hours of recess (max) per week, and POSSIBLY is allowed to talk at lunch for 15-20 minutes -- might be in soccer or some other activity -- I'd say my K kiddo's social opportunities stack up well against that.
There is this built-in cultural bias here that 'socialization' means 'spending time exclusively with a group of 20-30 kids the same age as you.' And that's not realistic. It's not even normal for our culture, let alone our species; we only started doing it historically here this past century. My grandma and even my mother attended one-room school houses with mixed grades of students (mom only for a few years). This is a recent phenomena and it's not the way humans spent most of their existence. Frankly, for much of our history, kids grew up socializing in groups of cousins or fictive kin, with 2-4 at most in age groups (infant, toddler, young child, child, adolescent). This from my perspective as an anthropologist. So, this 'ton o'kids the same age' set-up is recent, it has not basis in our longterm development. And it's pretty simple to provide one's kids with that exposure without having to put them in school. And once they're out of school, they will never again be in an environment were everyone they're with is within the same 1-4 years of age. Not at work, not at church, not at anything. So it's not necessary.