My guess for the college admission situation is at least partially because, until recently, a lot of public and private colleges essentially refused to recognize homeschooling as a way to get an education, so your chance of getting admitted was almost zero at a ton of schools. This lead a lot of people (myself included) to essentially join something that's legally an accredited private school, but functionally homeschooling, but you take your exams at a central place at a central time. For statistical purposes, I was private schooled, because that was what I needed to do to make it into college. Actually fully homeschooled people went to community college and then tried and often failed to transfer.

For reference I had to take several SAT subject tests to get into college ~20 years ago, the homeschooling co-op I was a part of did get accredited while I was attending so I think those tests weren't a requirement for other kids who went through an accredited co-op.