Could Paul McCartney sing his own Songs?

It’s not the bar band that has to pay. Any business that offers music as part of its package of entertainment, whether it’s live or recorded music, has to pay royalties. There’s an exception for a place that is playing music on stereo equipment that’s no more elaborate than someone might have at home.

This is correct. I believe bars that have live music must be a member of ASCAP (American Society of Composers, Authors and Publishers) in order to legally have cover bands perform without running into any copyright violations.