While I agree with y'all that genius is not limited to members of the upper class, what is particularly striking about the idea that the "Man from Stratford" was the author of 38 plays, 150-something sonnets, and at least two lyric poems is that once Shakespeare returned to Stratford from his time in London, he seems to have completely forsaken any life of the mind (he didn't leave behind any books or manuscripts in his will, and there is no record of him ever sending a letter to anyone).
There are many examples of people from humble circumstances educating themselves (Nathaniel Bowditch, Robert Burns, Benjamin Franklin, etc.). But not only does the historical record typically preserve anecdotes of these precocious youths borrowing books from wealthy neighbors, etc. they also, without exception, retained their love of learning throughout their lives. What is suspicious to me about Shakespeare is the idea that he somehow came from illiterate parents to be the single most literate person in the history of the English-speaking peoples, and completely turned his back on that life to live as a functional illiterate after reaching his forties.
I don't really have a "dog in the hunt" on the authorship question. But I'm open-minded enough to accept that there are enough oddities in Shakespeare's biography to at least leave the door open for another author.