Homeschooling comes up fairly often on HN, usually eliciting the same kinds of comments. Commenters repeat the same old tropes, generalize from anecdotes, and project individual experiences onto other people.
I raised and homeschooled three children. One of my daughters homeschooled from second grade to the time she started college. My other two children decided to attend public school at various ages, and then mixed that with homeschooling because I didn't force them to attend school.
Infants have obvious personality tendencies. Some seem curious and outgoing, others afraid and nervous. Some babies seem happy and comfortable with other people, and some recoil and cry when approached. They don't start as blank slates. The family environment, siblings, peers, and the school environment can exacerbate innate tendencies or push children to change. A shy child may withdraw and develop social anxiety at school, or turn more extroverted and confident. Personality traits change throughout our lives, and that happens faster and often more dramatically in young children.
"School," "homeschooling," and "socialization" don't describe specific or uniform experiences, though people use those terms as if they do. Some children have loving and nurturing home environments, some suffer abuse. Some children attend well-funded and staffed schools, others spend their years in a prison-like environment with the emphasis on crowd control. Some teachers have a talent for teaching and subject mastery, they can inspire children to learn. Others seem resentful, incompetent, disengaged, even cruel. General statements about public schools, teachers, and the process of socialization don't mean anything because of huge variations across broad spectrums in multiple dimensions.
When asked, teachers will cite parental engagement as the most important determinant of student success. Many, probably most, parents use school (starting with infant daycare) as a place to park their kids while they work, trusting strangers they rarely meet to raise their children. When you watch a classroom of children interact with the teacher and each other you can pick out the kids who have involved parents and those who don't. Of course most parents have no choice for economic reasons. Homeschooled kids tend to come from two-parent middle (I'd say upper middle) class families for that reason -- few families can afford to have one or both parents lose so much work time. Several studies of homeschooled children implicate family socio-economic status and income in the success of the children, already advantaged compared to the majority of public school kids.
Parents take a wide range of approaches to raising children, generally with little or no training or preparation. You have a child and suddenly you have complete responsibility for another human being, making decisions on the fly, subjected to conflicting advice and guidance. That other human being has their own personality, will, requirements, and eventually desires and opinions, and you have to discover those and adapt to them because children don't come with labels or documentation. Some parents pay little attention to their kids, often because the parents have too many economic, relationship, or other problems of their own. Some parents treat their kids as extensions of themselves, a chance for a do-over, version 2.0, and will start prepping their child for the future the parent wishes they had regardless of the child's inclinations and desires. Some parents want to control their children, usually in the name of protecting their child from the world. Parents who don't recognize and respect their children as separate individuals can do a lot of damage, both in socialization and in academics, and you see the results with both schooled and homeschooled children.
In fifteen years of homeschooling my own children I spent a lot of time with other homeschooling parents. The terms "homeschooling parent" and "homeschooler" refer to such a broad spectrum of motivations and approaches that they doesn't usefully describe anything at all. Some parents want to raise their children in a faith tradition (I met fundamentalist Christian parents, Mormons, Muslims, Jews). Some parents want to raise their children with no faith tradition (secular homeschoolers), but then push their own woo-woo and new-agey beliefs on their kids (opposition to vaccines, mindfulness, astrology, homeopathy, "Indigo children," etc.). Both religious and secular homeschoolers can indoctrinate and control their children, and put their own beliefs, desires, frustrations, and fantasies into their kids' heads. I call that another form of not respecting the child as an individual.
The socialization topic comes up so much when talking about homeschooling that I have to say something about it. Whether schooled or homeschooled, children get exposed to adults, other children, and peers their own age. Each child will react and adapt to their specific circumstances according to their personality. Nothing inherently prevents homeschooled kids from enjoying a rich social life, but parents often restrict social activities, and with whom their kids socialize. Nothing inherent about the school environment magically "socializes" kids. Some children thrive in the school environment, others graduate or drop out with emotional trauma. Some schools offer more stable and safe environments than others, largely a function of ZIP Code in the USA.
Homeschooling at its best takes parental engagement to the extreme: parents take complete control over, and responsibility for, their child's education both academically and socially. Parents might get good results from the right schools if they engage with the teachers and pick the right school (thus the popularity of private schools). Parents who recognize and respect the individuality of their children, and don't treat them as clones or property, who actively take part in educating their child (whether at home or school), have a good chance at raising a well-adjusted and functional adult. The less the parents engage and take responsibility, the less the parents respect their children as discrete individuals, the more likely they will raise a child with academic failings and maladjusted personalities.
Look at the results all around us, the products of public schooling and disengaged parenting. Homeschooled kids may have the same problems, but in my experience that happens less often, especially if you exclude the large number of religious fundamentalist homeschoolers in America.
Specifically for the HN crowd (to which I belong)... people who work in the software profession that cannot measure what we call productivity, a profession that cannot agree on "best practices" around even trivial things like indenting, should step back and take a more humble approach to opining and lecturing about parenting, education, and homeschooling. No one knows the right or best way to raise children, or how to "properly" educate them. Instead we have a lot of opinions and traditions and government-imposed rules, some useful and some bullshit. As parents we have to sift through all of that and make decisions on the fly that can dramatically affect our children for the rest of their lives. We can't refactor away a bully, a cruel teacher, neglect, or the effects of expecting an iPad to substitute for attention and care.