Actually, I believe it's literature that is the province of the writer.
Theatre, on the other hand, can only be realized as a collaborative effort. Yes, plays do exist on the printed page and for that, there are other awards given -- the Pulitzer, for one. But the Tonys are given for the *production* of a play and for that reason, while I agree the writer should be recognized and given a chance to speak (which inexcusably, doesn't always happen), it's entirely valid and appropriate for the producer(s) to be onstage and to accept the award in recognition of his/her responsibility for making that production happen altogether.
"No matter how much you want the part, never let 'em see you sweat." -- Old Dry Idea commercial