Amusing thread.
Think I'm with Jerry R. on this one. Maybe it's a bit like acting. How much does the accent have to do with the story or the character? The simple answer would be: depends on the plot and the character. Hamlet or Macbeth can be played in English, American, German, Russian, Afro-Caribbean, Japanese or any other accent. Willy Loman, I think, needs to sound like he's from New York.
Some songs, like some plays, have such a strong sense of place and time that you have to address the language in some way. If you can't then perhaps it's best to walk away from the song (or part). But then there are songs like 'Carrickfergus' which name a specific place but are about universal human emotions (in this case, homesickness). Anyone can relate to them and anyone is entitled to sing them.
On the other hand, I read an article in an Irish magazine a few years ago about the Waterboys which talked, amongst other things, about 'dabbling with identities'. Not a good idea. Wish I still had the article.