No matter how good Shakespeare is, either in absolute terms or just compared to his contemporaries, the fact that the First Folio was created so quickly after his death, and in such reasonably good form, and that there were many other sources also available for study, was invaluable in cementing him as the preeminent voice of that era.
We might find Shakespeare’s language stilted now, but other playwrights were worse then. The point is that it absolutely wasn’t florid or overblown (like other playwrights of the time) but more reflective of how people actually spoke then. Audiences then, like now, expected the language to not be exactly what they’d say to their neighbor. And Shakespeare reflected that better than anyone, particularly in the speech of his “dregs” of society.
Rather like how no one now speaks like Joss Whedon’s or Kevin Smith’s characters, but some of us wish we could.
I wonder how much Shakespeare Tolstoy actually watched. (Like Peyote Coyote says, it absolutely should be seen whenever possible, read only when necessary). And I wonder how good his translations were.