Additionally your argument even per-supposes non-incidental socialization. For example, my brother home schools his kids but they are rarely ever "at home." For example last week looked something like this for them:
--Monday they attended a city library survey of Shakespearean tragedies. Then they attended the community chess club that evening.
--Tuesday the university museum of modern history has a weekly K-12 program open to both public and home schooled students. They go almost ever week.
--Wednesday was the robotics club at the community center. The home school association soccer program. And a STEM program (they are doing game development and working on their robot for the club competition) through a local STEM academy.
--Thursday is back to the university where a number of the individual colleges have courses for home school students on history, science, and math (my understanding is that these programs are open to anyone, but home schooled students make the largest block of attendees, followed by elderly retirees.)
--Fridays are at home for papers, research, and computer classes.
I guess I am saying that your perception of home schooling might not be a broad as it could be.