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 is a big step up from $0$-categories and on the other hand it is a low enough category level that one does not need to worry too much 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 one can attempt to ignore it by strictifying, and at the $3$-categorical level one cannot even always strictify..

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..