I've heard that the controversy over Shakespeare's plays authorship was started in the 19th century, by educated "elitists" who were insulted by the idea that a man who hadn't been to college could produce the body of plays attributed to him.
And I do agree that playwriting wasn't considered that high class of a deal during Shakespeare's lifetime. It was a way to make a living, it was popular entertainment, and it wasn't even intended to be published.