Billy you're too harsh!
My take on this is that you need to feel a song rather than believe it. And by the same token I believe that as a singer, by getting inside the song, you can get a taste of what it would be like to believe whatever variant of truth is embedded in the song.
Like an actor, you can suspend your disbelief and for a brief space of time empathise with what you find in the song.
So then it would be a good exercise in understanding for an atheist to sing a faith song & for a believer to sing an atheist song.

PS. I've never sung Will the Circle be Unbroken - but I'm very sincere whenever I sing Les Barker's Will the Turtle be Unbroken.......