I have a very weird perspective because both my parents worked from home AND homeschooled us. We did a lot of church activities and swim team. We were 0% lonely, in fact I longed to be alone as a kid ;)
And I was homeschooled by a single mom who obviously had to also work to put food on the table and a roof over our head. I'd say I was 95% lonely. In general kids need the socialization an organized school would provide.
Wow that must have been difficult. I would personally not homeschool my own kids, but I think it's a good option in certain situations. I'd like to send my own kids to a school, but one that's maybe more a "hippie" school that allows them ample free time and activity.