Fridge Brilliance: Of course Yukio didn't forsee Yashida's death, because he wasn't really dead.
To be fair, she didn't foresee him actually dying at the end of the movie either.
You know what, we never see his dead body after he gets thrown out the window by Logan. Maybe he's still alive?
Yashida keeps calling Logan "kuzuri" at his sickbed before his fake death. While "kuzuri" is allusion to a mythological japanese creature with huge claws, it also sounds a lot like "kusuri" or medicine. Guess "who" is the medicine that Yashida plans to obtain to become immortal.
You know how Logan acted like he didn't remember anything about Yashida and never went back to get that sword? Well that's because he didn't remember. He had bone claws in the flashback segments, meaning he hadn't lost his memories due to the adamantium bullets yet. Though there's still the Fridge Logic of why he never went back to get it after leaving his squad...