If you’re wondering how many times the traveling coin 360s you have to account for the fact that it is rotating both because it’s rolling and because it’s going around the the other thing. You can simplify this by just imagining it’s the edge again, but the edge is going along the radius of the inner and traveling coin (3+1).
Dumb it down. How far does a point travel? X units. X units / coin perimeter = coin rotations required to travel X. Ergo the only question is whether the distance traveled is along the perimeter of the 3r coin or if it’s based on the center of the traveling coin (3+1).
Seeing the turns in two axes made it clear to me that the forwards rolling is always the length of the track because changing the shape doesn't change the length, and the sideways turn is always one for any simple loop or it wouldn't be a circuit. That fixes your question "how each diameter of distance traveled by the disk center equals one rotation regardless of the shape of path followed" - you say it's obvious for a straight line, and curving the line can't change the length of the line, the extra rotation to go round a circuit must be one, so there's nothing which can vary.
(If you lie the unicycle down on its side, the 1 complete turn to the right needed to follow the curve around to the start doesn't go away, it has to happen, and now it has to happen in the same axis as the rolling forwards happens so it's much less clear (to me). If the only thing the wheel can do is roll forwards, this extra movement from earlier to avoid getting away from the track can only be extra rolling - the extra one turn for the loop, spread over the N turns for the distance).