You should increase your category level if you think it'll help you understand something you're thinking about. Otherwise, don't.
To the extent that there's something special about $1$-categories, it might be that on the one hand it's a big step up from $0$-categories and on the other hand it's a low enough category level that you don't need to worry about coherence conditions. Already at the $2$-categorical level an important distinction arises between weak and strict $2$-categories (when I say "$2$-category" I mean the weak thing by default because the alternative does not generalize to higher values of $2$) that takes some time to appreciate, although you can attempt to ignore it by strictifying, and at the $3$-categorical level you can't even always strictify..