Thanks, MandolinPicker, for that link. It explains why it's usually
not necessary to get the copyright holder's explicit permission to use a song. That's usually covered under a previously-negotiated blanket license, not unlike that of a radio station. At least, as long as the music is incidental and not synchronized to the action on the stage.
That doesn't mean there isn't room for lawsuits, though. But
in most cases, when legal action has been pursued, the artist has lost. Most of the time, the artist does not sue, preferring to argue their case in the court of public opinion. (After all, even a wealthy pop star is no match for the bottomless resources of a major political party where the majority of candidates are lawyers.)
Interestingly, when the same scenario occurred in France, the political party paid a substantial out-of-court settlement. Different standards in the EU, I suspect.