I have a theory for why Dog has a fear of being useless to others:

There was a situation where someone's life was at stake (or partially at stake), and Dog couldn't save them. Maybe that person was saved by someone else. Maybe they weren't, and were doomed from the start. Regardless, Dog felt very awful for not being able to do anything, and thus gained some PTSD when the Wizard turned him into a figurine.*

I'm assuming a very similar thing is going to appear later on in the series, with Dog's usefulness being tested the entire time it happens.

*The original version of Cat Bird Dog had the Wizard turn the imprisoned villagers into plushies, while in the new Webtoons version, they were turned into figurines.