They can cover/play their own material as long as they are registered as the writer and/or publisher, but no one else’s work unless they have permission. Membership only gets you royalties on works you own.
It is still owner’s rights regardless of who owns the recorded material or the creation of the material. You understand that most artists do not own the publishing or even the recording rights, but most of them get permission to perform, and maybe even use some of the recorded material. Managers and Labels usually screw the artists for those rights. Look into John Fogerty’s nightmare with his manager. He couldn’t play any Creedence tunes for years. Same thing happened to Springsteen and many others.
BTW, Fogerty and Springsteen both won their rights after years in the courts.