There are, as I see it, five point-of-view sets in songs, and some of them present no problem at all to a singer of either sex, and some perhaps more:

There's what novelists sometimes refer to as "the eye of God" point of view. It's told completely by what might be called an unidentified outside narrator.   There's no problem here.

There's the song told entirely in the first person by a female protagonist. This is one which might just be tricky or uncomfortable for a man to sing. Or perhaps it may be comic for that reason.

Then the converse is the song in the first person by a male protagonist.   Perhaps a woman might have problems with this, but somehow I don't think that women generally would have as much discomfort with this kind of song as some men might with the female song above. I wouldn't think a "male" song sung by a female singer is so likely to be made comic as the other way around.

Then there's the "extended quote" song, as was pointed out earlier about The Frozen Logger. Another example would be The Bold Fenian Men. Actually, in neither of those two songs is the sex of the initial narrator specified, but I think most of us would think of it as male.

One of the frequent characteristics of the traditional ballad is the unanounced change of "speaker" or point of view. A good example is the ghost version of The Grey Cock, which freely changes narrators. Or maybe I should say there is no "narrator" as such, but it's told in a combination of "eye of God" and unannounced quotes from both parties, which it's up to the listener to sort out by context.   This kind of song should present no problem to a singer.

Dave Oesterreich