I used to think the same, but ASCAP has a very nice, easy to understand page about licensing for political events that is super informative.

I posted this up a level, but being as you seemed to have a better understanding about this than most other commenters, I wanted to post this as a reply to you too so you would see it.

If the campaign events are properly licensed, can the campaign still be criticized or even sued by an artist for playing their song at an event?

Yes. If an artist is concerned that their music has been associated with a political campaign, he or she may be able to take legal action even if the campaign has the appropriate performance licenses. The campaign could potentially be in violation of other laws, unrelated to music licensing:

The artist’s Right of Publicity, which in many states provides image protection for famous people or artists The Lanham Act, which covers confusion or dilution of a trademark (such as a band or artist name) through its unauthorized use False Endorsement, where use of the artist’s identifying work implies that the artist supports a product or candidate

As a general rule, a campaign should be aware that, in most cases, the more closely a song is tied to the “image” or message of the campaign, the more likely it is that the recording artist or songwriter of the song could object to the song’s usage by the campaign.