I'm not familiar with Osomatsu, so I could 100% wrong but doesn't Kanzaki's lines about how "the story won't progress without me" simply means that without him, there was no way for Mogari to see the ghost?
I don't think it's outright confirmed, but in chapter 1, Mogari said "I can see them, they're on my turf now." which implies that Mogari can't exorcise spirits or can't exorcise them effectively if he can't see them. Without Kanzaki coming along, it might mean that Mogari won't be able to exorcise the ghost that's trying to kill Tamon, hence his 4th wall break line about the story not being able to progress without him.
Anyways, you still did a phenomenal job translating this, kudos!!