So for people who wonder about the finger bone...
It's already mentioned by Saito in the manga, but it was necessary for Akechi Mitsuhide to recover Nobunaga's body (and head) for his rebellion to be justified, and in real history he failed to do so because Nobunaga burned down the temple specifically to prevent such recovery. However, after the fire was extinguished some bodies were likely recovered from the ruins but were probably impossible to identify.
Traditionally, unidentified bodies of victims in such a brutal incident would have been buried ceremonially not in a cemetery but in a special shrine. However, there are no records of what had happened to the bodies from the Honno-ji incident.
In Japan, therefore, where Nobunaga's body disappeared to is a major historical mystery. This is a mystery Ken's father had been trying to solve, and probably did manage to in the original timeline. He had found Nobunaga's finger bone. This is the item he'd left behind with Mocchan.
So what is the importance of the finger bone? Well, the manga suggests that the discovery of the final resting place of Nobunaga's remains was what had triggered the time travel episode, starting with the earthquakes that eventually destroyed the hotel. The manga repeatedly stated that Nobunaga's fate would be determined only by the heavens OR NOBUNAGA HIMSELF. The time travel event is hinted to be the latter. That's why, like Akechi said, heaven wept when its will was defeated.
The finger bone was the catalyst that pulled all the future people to Nobunaga's time to exercise his will and cancel Honno-ji. Probably Ken's dad, a man who knew history in detail and had the adventurous spirit (and obsession with Nobunaga) to try to change it, had been the intended target, but because he'd handed over the finger bone before the event, Nobunaga's finger bone ended up with four chefs and a maitre'd instead.
The finger bone literally sent Mochizuki off on his adventure to become the second pebble meddling with the boulder of history, allowing history to be changed completely.
The finger bone being discarded closes the future path to our original timeline. The wish of the fallen Nobunaga had been fulfilled and the timeline had been "corrected" from his perspective. The catalyst was therefore no longer needed.
