Depends upon the function of song. There are nonsense songs for fun and entertainment. For those, you wouldn't want "sense" as much as cleverness which is easily comprehended by all.

I think people have lost all sense of the different types of songs there are. Narrative songs tell stories, lyric songs communicate impressionistically. But all songs are an attempt to communicate, so yes, they do need to "make sense" to the listeners.

I think the difficulty some have is cultural. What makes sense to one culture doesn't necessarily make sense to another culture. So one's ethnic/ancestral upbringing and living environment as an adult makes a lot of difference.

If one is heavily immersed in mainstream Anglo culture, the cultural references to a traditional culture from Europe or Asia or Africa which isn't your own, often doesn't make sense. And vice versa, of course.

I've seen a correlation between the best contemporary singer-songwriters who compose mostly lyric songs, and who also paint. They seem to be able to create strong imagery which helps people "see" the metaphors in the lyric.

But that's probably just me.