That is not the case with these boys.
One of them was in school through 3rd grade and doing ok. He wasn't a star pupil, but he made friends, expressed his opinions, made decisions on the small stuff and was interested in the world around him. He was sociable, but needed some time to warm up to both the situation and the people around him. My 71 year old husband is like that, and he does not suffer from severe anxiety.
Once home schooled, his parents, despite saying that they would get the kids involved with teams, projects, etc. outside of school, did nothing to encourage any activity outside of the home. They always had an excuse for not doing so - most of them amounted to "I don't want to". They actively encourage the fear of school and people by threatening to send them to public school. I don't think the parents have a clue about how devastating their overall attitude has been to the boys. Unfortunately, the parents are 'all about me' types. They will never understand what they have done to those boys.
The first year of home schooling neither boy was doing any of the work because neither parent could be bothered. Mom got sick and had surgery so I went out to help out. The only reason the kids got started on the school work is that i made it my priority to get them going. The shy one was so eager to learn that it astounded me. He managed the entire school years worth of work from end of January to June 1. That was after telling me he had short term memory problems - diagnosed by a so-called learning disability expert. My response to that was, "Me, too. But I know how to help you with that." No one ever just sat down with him to explain how to study. He didn't understand why text was bolded, italicized, in tables, etc. Once encouraged, he was amazed at what he could do. So was I.
The younger one, who has Aspergers, is very bright. He told me he didn't have to do school and he was smarter than me. His mother encouraged that feeling in him. I dealt with it through concrete examples. For instance, he could do amazing math on a calculater. So I wrote out a couple of 3 digit addition/subtraction problems and asked him to solve them. He had no idea what to do. A day and a half later, he could actually do addition and subtraction. He is smart!
His temper tantrums ran the home, so I gently disciplined him - it took 1.5 days to modify his behavior with me. I have no idea how he behaves at home now, but when he is with me he's a great kid.
Parents of a home schooled child make a huge difference in the success of the home schooling.