I don’t know who else could pull this off. Jack White is the definition of cool. He doesn’t worry about the gender problem with this Dolly Parton song. He just lets it fly.
Come to think of it, Bob Dylan sang “House of the Rising Sun” — a Dave Van Ronk arrangement — from the point of view of a girl who’s fallen tragically into prostitution. So singers inhabiting a character of another gender certainly isn’t unprecedented. There are surely many other examples.