Technically, Picard and Riker are on a ship that isn't theirs. They're "borrowing" it. It's not theirs. So I see the argument of the TNG characters having no connection to the ship (other than it's called "the Titan") as a non-starter. The fact that it's not their ship and not what we associate with their ship is the whole point.
The whole point is that the only way they could get a ship to “borrow” was that Riker had a personal connection to it. This is Schrödinger’s character beat, is it? Picard and Riker are so out of the loop that they have no ship of their own, except this ship where Riker’s personal connection is so strong he can just tag along for a joyride whenever he feels like it, but he has no connection to it because the characters’ day is over and they’re on their own and have no resources, and so on. It’s his old ship and not his old ship at the same time.