Almost none of the artists or after market owners directly license the music. It’s almost always done through a group like ASCAP. That said, I don’t know if use by political campaigns and conventions is considered fair use or is done by license. Either way, the leftist artists who spout off about conservatives using their music really don’t have a say in it.
Yeah they do because ASCAP and BMI licenses have an exemption for political events. The artist could sue, but unless the offender keeps using the song over and over then there really isn't any point. So Trump plays the song once, they act all outraged, they get their name in the news, Trump never plays the song again, and it's all over.