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.
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.
And so on.
The fact that he was a businessman demonstrates that copyright isn't completely necessary to make creative works a viable business.
Like that never happens now. How many independent creators did successfully sue a studio since copyright law has come into play? How many pitched a script to a studio, only to be rejected, and having the same idea rewritten on a spec by others?
Is it worth it to keep Steamboat Willie under lock and key?
Might as well tax the IP secondary sales and just give every living author with works registered in the Library of Congress a stipend.
Snark aside, I think there are people who make a living creating things that don't rely on copyright (music artists that make their living off merch and live shows, theatre as mentioned above), and even if there weren't, that's not particularly surprising because copyright laws exist - without them, there would be stronger incentives to discover other revenue streams.
Theatres remount things and profit from it. Theatres do original works without much intention of selling the rights later and profit from it. Obviously there are going to be a ton of examples where copyright is used to generate revenue, because it exists and why wouldn't you copyright something when there's no cost to doing it and potential upside. But there's plenty of evidence that live theatre would still exist without copyright.
Broadway might still exist. All the theaters, especially community theaters, across the country that don't have Eugene O'Neill or Richard Rodgers would have a terrible time pulling a full house if Bob's nephew wrote all the plays. If there was no way to protect the works made for Broadway, the incentive for people to spend the time writing, composing, arranging, designing sets, making costumes, and choreographing those shows largely disappears. The show can be redone down the street or in Omaha the next week, and all the money goes to the theater doing the ripping off. The show can be videotaped and shown on Hulu the next night, and all the money goes to Hulu.
Shakespeare's plays are beyond the time limits on copyright, which there definitely should be. They are also considered classics. The performance, believe it or not, can still be under copyright. You can't generally go in and record the performance and put out your own stream or DVD of it.
Copyright didn't exist forever, yet we have music and books from before then. Shakespeare was before copyright. Most of the best creatives create for the joy of creation, not for money. Like open source.