There are plenty of traditional songs which are primarily lyrical, where the "story of sorts" is secondary.

For example, Carrickfergus certainly isn't a slushy love song, but there's no clear narrative in it. It's made up largely of elements which have been used in ballads, but that doesn't mean it's some kind of defective ballad. It doesn't set out to tell a story, but rather to present a mood. That's what I mean by lyrical. And there's many another example of this kind of thing.

There are storytelling traditions where teh story itself is not in verse, but where at some momeents of the narrative a song will take over. Those songs aren't ballads.