Looked up this guy and organization. Makes you wonder what influence the procedures of ATI had on the situation. I don't think their "home church" was under its umbrella or authority. I've heard JBD say on the show it follows the traditions and practices of the Southern Baptists, of which the elder Duggar was raised in. Most of my family is SB and I don't see hardly any connection between their beliefs and practices, especially the Quiverfull and ATI materials.
Here's an article written a year ago by a former ATI student and teacher-
http://www.huffingtonpost.com/micah-j-murray/growing-up-in-bill-gothar_b_5228166.html
It might shed some light on why they took this course of action with their eldest son. The graphic in the article about handling abuse is disturbing to say the least. It appears to be addressed to young women who are to blame on some level for the sexual abuse. You're right Bill Gothard (never heard of him until today) was a manipulative, nasty person. Brain washed a lot of families.
Again, as someone stated, there might not be any connection at all but I do see some possible dangerous influences in the Duggar's life that could have influenced them to handle the situation so poorly. Especially if they followed the graphic/procedure in the ATI materials.