A wonderful (and compelling) article.
I certainly agree that it is reductive, to say the least, to refer to Shakespeare as a great story-teller. His texts are remarkably powerful, but they are not powerful because they tell wonderful stories. The texts do not, of course, tell stories at all: they equip the actors and actresses with tools of wondrous dramatic power. There is a story (or many stories) to be found in each of them, but the audience is not “told” what the story is: it must find a story itself, and it finds it in the dramatic performances it experiences. It is hard, in fact, for me to think of a way to more completely miss the mark as to what it is about Shakespeare that is so compelling than to say that he was a great story-teller.
And if watching many modern plays does not involve substantially different experiences than you would have if you read the scripts of those plays to yourself, then something has gone very, very wrong.