I say that because I'm a homeschooled child who is now an adult. From my immersion and circle of friends who were also homeschooled, these are my takeaways:
Parents are untrustworthy proxies for their children's experiences. If you surveyed parents, they would all say that they kept their child socialized through homeschool groups. If you surveyed the children, they would strongly disagree. The point made in the ask HN "and have witnessed it to be very social" is the same one every homeschooling parent I've ever met makes, and pretty much every homeschooling child disagrees with. (And I've met a lot of homeschooling parents!)
An astonishing number of homeschooled acquaintances I know have estranged themselves from their parents, and would cite their decision to homeschool as a factor in that decision (> 50%, n=~25.) Of the 5 children in my family, 4 cite homeschooling as a lingering negative that affects their social relationships to this day. 3 have forgiven my parents for it. The others still limit contact with them. (Siblings are all around their 30s.)
Children who are still school aged are incredibly unwilling to insinuate that homeschooling doesn't work for fear of the consequences. If you see your parents devote their time to homeschooling you and are constantly saying stuff like "You'd think that homeschooling isn't social, but look! We found 5 other weird families that our children only pretend to like because they're desperate for a peer group, and they get along OK!" you would also be hesitant to say to your parents that they approach they've dedicated their life to makes you lonely.
This is a decision, that, if you mess it up, you really mess it up. Be cautious interpreting a study that only asks parents their opinions, given by a consultant with a self-admitted homeschooling bias.
For the subjects she studies at home, she's been able to go at her own, faster pace and do in-depth projects, like writing an 80,000 word novel for English, which made learning both fun and meaningful. At the end of last year she already had five APs, all with 5, and she recently sat for the SAT and got a 1570. I think every child is different, but for mine, homeschool has simply been incredibly good for her, and loneliness wasn't a big factor.
* Parents with multiple children don't necessarily have them all in the same school/homeschool situation.
* One child might be lonely and another child not.
Others have mentioned the problem of grouping children together into the same sentiment, so I won't comment on that. But I feel this survey suffers from some priming and conformity bias issues.
The survey is very "soft" in the sense of asking opinion questions, which can vary wildly with the respondent's mood, time of day, time of year (just had Christmas together as a family and everything's good), etc etc. A single survey at a point in time has basically no value, which is why I generally disregard surveys I see in the media. Surveys really get most of their value through asking the same questions repeatedly at different points in time so you can "average out" a lot of that bias. But in order to have results with any legitimacy, you almost need to split each question that you actually want to ask out into several different questions, some of which have numbers attached to them.
For example, the question "Does your child/children have friends (not including members of their immediate household)?":
A parent is more inclined to affirm their biases and assuage any guilt they might have over their child's loneliness than to help you get accurate survey data. It also depends on what a parent considers a friend versus what the child might consider a friend (someone to hang out with vs someone they feel a real connection with and are comfortable sharing things with, for example). I'm guessing you'd get very different answers if you asked the child the same question.
I can't imagine a parent is very willing to say their child has no friends at all, but you might split the "core" question out into multiple different questions - "how many close friends does your child have," "how many acquaintances," etc, with answers in different brackets (1 or fewer, etc). And then ask the general sentiment question after. The advantage of this is that you actually have something to compare across time. If you only have sentiment questions, it's almost impossible to know what the respondent has in mind for a "friend" when they answer the question.
I have the same feedback for the "How often does your child or children complain of feeling lonely or not having enough friends?" question. Different personalities will complain more or less, but that doesn't indicate how lonely a child actually feels.
A minor point, but without randomizing the question order or randomly flipping the answer options (ordering from negative sentiment to positive sentiment instead of positive to negative), you're very prone to influencing answers by "leading" the respondent to certain answers. You'd be surprised how much of a difference this makes.
Writing good surveys is very difficult, but you might pull up a list of common survey biases and run each question through the list. My 2c.