Fuu Is the Reason There Is a Story at All
SPOILER!
Mugen and Jin would happily keep fighting until one of them dies. Fuu interrupts that ending and makes them promise to help her find the samurai who smells of sunflowers.
Why it matters
She is not the weak third member attached to two swordsmen. She creates the goal, keeps the group moving and repeatedly forces two men built around independence to remain responsible to somebody else.
Episodes 1–4 Turn a Death Match Into a Traveling Arrangement
SPOILER!
The tea-house duel establishes
Mugen and Jin as rivals. The yakuza conflict immediately tests how fragile Fuu's bargain really is when the two men take jobs on opposite sides.
What changed
They can split, fight and complain about the promise, but they have already begun behaving like people whose choices affect the other two.
Hunger Is the Most Reliable Fourth Member of the Group
SPOILER!
The trio repeatedly takes terrible jobs, gambles, chases rewards and accepts suspicious opportunities because they need food and travel money.
Why it matters
The road trip stays grounded because survival is usually mundane before it becomes dramatic. History enters through empty stomachs as often as through swords.