Started thinking as I read your post, isn't " fulfilling some purpose in a society, like a gearwheel" and "have a distinct, important role in their pack" the same thing? and isn't that why dogs fit so good in human packs.
No, because the gearwheel can't turn on it's own and is just to be replaced if it's defective.
The role in the pack can change on agreement with the pack, and members bring themselves in as the individuals they are.
Or put another, more simple way:
Function can easily be replaced, but not personality.
Dogs integrate well with humans, because they see no big difference between human and dog. The dog becomes a little like the human, which helps building relations.
Relationships wouldn't be possible without adjusting to each other to some extent.
I still feel like i didn't make the point clear enough, but it might give you a good hint.
Sure, there are "functional dogs" too, but these were not inside the scope of my example.