Musicians already earn a large portion of their income from things other than sales of their albums - whether it's touring / merch / advertising deals. All of that could exist in a world without copyright. The equivalent might not exist yet for movies, but in a world where movies were freely distributable, someone would find a way to still make it a business. Maybe the giant blockbusters would go away, but maybe that's an acceptable price to pay.
But if there was no copyright, why did the Globe owners pay him at all? Just perform the old plays for free! He was payed for creating something new, copyright was worthless because we did not live off selling old works.
Copyright couldn't help Shakespeare to create more or better plays anyway, he was limited by how much boot-licking the Queen was acceptable to the audience, and how many "obscene" stuff he could sneak in without angering the Queen or the Church.
Shakespeare made a living from the performances of his plays, not the written documents. Musicians made money from performances too. Most famous historical painters and composers had wealthy benefactors that enabled their work.
Creative people obviously enjoy creating, but professionals are doing it to earn a living. Look at the books on your bookshelf and the movies and TV shows you watch. How many were made purely for joy with no expectation of making money? Maybe some, but not most.