Gothard has absolutely nothing to do with this case but I can see why the Duggars might get involved. I'm not a fan of the Duggars or their right-wing moral and political agenda but they do have a fan base and a case like the Stanleys will grow it. The Duggars not only live in Arkansas, the Duggars are very active politically. Jim Bob Duggar has served in the legislature and ran for the Senate.
Hal Stanley is a retired Southern Baptist minister. There is nothing that suggests he is the leader of a cult or that his children are sexually abused. Corporal punishment is allowed in Arkansas public schools and that won't change because the majority of voters share the Christian belief that sparing the rod will spoil the child.
I can't think of a better way Josh Duggar can push forth the right-wing political and moral agenda of less government than to have a state government agency intrude, disrupt and divide an Arkansas family, not give a reason for it and hide behind a gag order.
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JMO