This goes back to the fundamental principles of copyright that have been globally held right back to the 19th century. There is a right of communication and a right of reproduction. Each of these rights has an individual value, and each of these rights is used differently. They pay for the performance of the music, on one hand, which is one use with one value. The right of reproduction is another use with another value.
They are not paying for the same thing twice; they are paying to do two separate functions, and they're paying for the ability to do each of those functions individually.