If the lyrics are great, I listen to the lyrics. If the tune is great, I listen to the tune. Best, of course, is when they are both great. I was first attracted to traditional folk music by the melodies, but quickly came to love the quirky lyrics. I can enjoy a great melody with bad lyrics better than I can a great set of lyrics with a bad melody. If the melody is uninteresting, I tend to just tune the whole thing out.
A bad performance or a bad mix of anything, vocally or instrumentally, is just bad. I think the OP was asking about the psychology of what grabs your attention rather than whether or not the performer is competent.
I think I differ from a lot of folkies in that if there is an accompaniment, I want to hear it. A standard folk music mix often has the instruments so far back in the mix that you can't really hear it, and it isn't contributing much to the overall sound of the piece. I'd rather it was a capella. A lot of rock music, on the other hand, has the vocals too lost in the instrumental mix. I like something somewhere in between.
I am an accompanist, and my experience is that the instrumental backdrop has a big influence on the overall experience of the song for the audience. I can play four different styles of accompaniment behind the same vocal track and make it sound like four different songs, each with different emotional contents.