Do pardon my arrogance here. These are merely ideas that more learned and rigorous thinkers might build on.
Not that I know much about Shakespeare, but I would still venture some thoughts on why he remains the outstanding playwright in Anglo-Saxon, or perhaps all theater. What exactly changed after his days that prevented the emergence of giants of equal stature?
The suppression of collective input
The Globe Theatre, if I understand correctly, was a fairly raucous place. It had no seating and people could come, go and drink at their leisure there. I trust that they were not slow to offer feedback on a performance and a text. Before a play went to the printer, it had been thrown to the wolves at the Globe and thoroughly chewed. The actors probably contributed their own enhancements. Yes, I know that the idea of actors upholstering their own parts has been derided vigorously over the centuries, but remember, these actors died on stage, if their added curlicues failed to win the audience. Later actors enjoyed a 'civilized' audience that held its mouth while the masters were at work and provided their snide remarks only afterward: No possibility of fine-tuning based on immediate responses to each phrase that an Elizabethan audience may have provided.
To provide an illustration of this, allow me to present a modern equivalent. Until the1930s or so, screen comedians would inevitably arise from the Vaudeville circuit, a huge network of informal theaters throughout America catering to a drunken audience that would boo performers off the stage if they failed to entertain within the first minute. Regardless of talent levels, comics like the Marx Brothers, Buster Keaton, Jack Benny, Fatty Arbuckle, Abbott & Costello, Laurel & Hardy, Charlie Chaplin, Fred Allen and even the Ritz Brothers had a sense of comedic timing hammered into them by these audiences that you will find woefully absent in their colleagues from later, canned laughter eras. Today's standup comedy circuit would help if it did not focus on single stars doing monologues. Which is why today's comedy films present these usually separately: The camera cuts to a single person who gets to star with his or her line and then to the next one. Little interaction. Nasty things, today's film comedies. Brrr. But back to the topic on hand.
Collective honing of individual genius
It does not seem accidental to me that our greatest authors, Homer and Shakespeare, are also the most anonymous. Homer's works were only put into writing after centuries of repetition and improvements based on audience reception and ease of memorizing by hundreds of professional bards.
Can you even imagine what that does to a text? Please, could a few hundred writers go over this essay and polish it up to the brilliance denied to it by my individual failings? No? You're not going to do my work for me?
How about this: We'll create a fictitious author who will reap the blame and fame for it and forget about Rene Kita who merely started the ball rolling? We could call him, for instance, William Shakespeare.
That would be a plausible explanation for the public invisibility of Mr. Shakespeare himself, even if he did exist as a person: If he had started putting on airs, his colleagues and friends (and possibly the wife, like the many 'proofreaders' humbly acknowledged for their "invaluable input" in thousands of forewords), who had actually cut and polished his rough diamonds into the masterpieces of the printed folios, would have drawn and quartered him.
Alas, no more
This how I imagine an Elizabethan audience: As soon as any single sentence failed to impress with profundity and well rounded tones, there would be fart sounds from the floor, conversations getting louder in the back and people wandering off in search of a drink. Anyone behaving even remotely like that in a modern theater would be ejected by the ushers.
Interestingly, there are still a few art forms that have grown out of environments like that, jazz in particular and, slightly differently, dance music with its audience that votes with its feet. And lo, jazz lost its innovative impetus as soon as the white middle class with its quiet, foot tapping respect for the artist arrived on the scene. Charlie Parker still had to fight for attention from inebriated gangsters and impatient jitterbugs with every note he played. Do tell me his art suffered from that.
Say, this turned into an apotheosis of rude spectators. So do feel free to point out my failings as rudely as you wish in the comments. It will not turn me into Shakespeare, but it might help a bit, anyway.
Oh, and do go on and rip this off and improve on it if you see anything of value. That is how culture lives and grows.
Addendum, as in oops
Should have mentioned that Shakespeare himself acted in his plays: He could experience word for word how each sentence worked on the audience. There must have been a lot of rewriting going on between each performance. That's how vaudeville acts honed their performance. By the time Who's On First made it to screen and radio, Abbott and Costello could do it in their sleep - I bet they were having nightmares about it by the end of their career. And it's a fine skit, not a single misplaced word.