ASCAP doesn’t cover political campaigns. Also, most campaigns don’t want to be seen performing an artist’s work against their direct wishes; it doesn’t tend to look good.
Many Republicans actually believe a lot of stuff that’s true. They just tend to have One Big Issue that only the Republican candidate will support, usually something bigotry related, but for a few it’s HCGW related.
The interesting bit to me here is about Woodward. He appeared to be advising certain clients in a way that was both illegal and which put the client paying the bills at an advantage.