As I posted above, language is fluid, but linguistic historians (or whatever the name du jour is) like categories. Early Modern English is a misnomer, and used primarily to teach people about the transition, but it has no place in the real world.
But you agree that there was a transition, that the language of Shakespeare was a part of the evolution of the language we know as Modern English. Indeed, Shakespeare coined or codified more of the vocabulary of Modern English than any other single writer (although the King James Bible was no doubt highly influential too). Granted the obvious fact that language is fluid and categories inexact conveniences, I stand by the assertion that if you have to use a category label, "Modern English" is more appropriate for Shakespeare than "Middle English."
I don't like labels for transitional things like that, but I suppose you have me there.
Now, about Jellico being a jerk???