It's a huge subject. The longer I HE, the more reasons I find to do it.
Just starting from the social angle, since that is what your partner mentioned, here are a few questions to consider:
Given that the main objection people usually make to HE is the notion that home educated kids will lack opportunities to have fun with other kids and learn social skills, I think it's interesting to question how successful school is at both of these aims. There's no doubt that some kids enjoy the social aspect of school and learn good social skills there. But that isn't an explicit objective of the education system and it's very hit and miss. There's also no doubt that many kids suffer socially at school and that it can teach kids social skills which don't serve them well elsewhere.
The lack of freedom at school restricts children's social choices, which can have unhealthy results. Is it desirable to socialise mainly with a single fixed group of children who are exactly the same age? Does this encourage unhealthy comparisons which can have an impact on peer pressure, exclusion and bullying? Why is bullying so much more widespread and difficult to resolve in schools than in other environments where children spend time together? What are the social consequences for children who are not average? What do their peers learn about how to treat people who are different?
What happens when people who must remain together all day every day need a break from each other? What do they do when they have a bit of a headache or are just fed up with being with other people and want some peace? How many adults would be willing to work in an office with so many people in it, and under such crowded conditions? If we wouldn't tolerate such an environment happily, how can children do so? How do people behave when they are under constant stress, even if it is only low-level stress? Does it encourage them to be kind and patient with one another? My younger daughter doesn't like being with crowds of people for hours on end. The more she is crowded in with people against her will, the less she enjoys their company and the worse she behaves toward them. What lessons would that teach her?
When children need help and coaching to resolve their differences, how much adult attention is available to them? When 20 to 30 children are sharing the time of just a single busy adult whose primary job is to educate them, and when there are no older children on hand to act as role models, how do they learn what to do when someone whacks you, or when you are feeling left out, or when you feel annoyed by somebody and don't want to play with her but you know she'll be lonely if you don't? With a tricky social challenge at school, who has the time and interest to observe what is really going on and discuss it with the children to help them find productive solutions?
What proportion of the school day is available for proper socialising, by which I mean playing and talking in an undirected way? Add up all the school playtimes and breaks - does it actually add up to more hours than you'd get from going to the park or having playdates a few afternoons a week? My older daughter used to spend solid days playing nonstop with her friends, not the odd half-hour here and there. These long stretches of time created a different quality of play and different type of social interaction.
I could go on. Oops, I think I did go on, LOL.