This is actually off. For the JK, T7 doesn't have any quests. Meanwhile the JK's second companion has two. I know off the top of my head that the JK's fifth companion(and I'm sure other companions as well) has a quest in the datamined files, but that quest is not in the game currently.
Fair enough. I guess JK falls into the "Very few". I know on all of my Empire characters, the story is around the first companion, except as I said, the BH, which has 1 later on for the 4th companion.
I suppose it makes sense, with the companion being a droid, that would change.