Wait, Is Richard "Selfish Gene" Dawkins a Shakespeare Denier?

Spotted the headline Richard Dawkins Criticizes Shakespeare for Lacking Elite Education this weekend and thought, "Aw man, here we go again.  And I liked Dawkins' work, too."  And by "liked" I mean, "Read The Selfish Gene in college."  I have no personal experience with his well-known views on atheism.
Well, here's the quote in question:

Sorry to be boringly predictable, but Shakespeare. Who are you? And how did a humble country boy like you become the greatest genius, and part creator, of our beloved English language. Might you have been even better if you’d studied at Oxford or Cambridge?

That's it?  Hmmm.
The Atlantic Wire article I'm linking to goes on to say stuff like, "It remains unclear what Dawkins could have meant by 'even better'."  Really?  How unclear is that?  Are we arguing now that Shakespeare truly achieved perfection and that to even suggest that he could have been better is sin? Would he had blotted a thousand!
Seriously, though, the article is attacking Dawkins' reference to Oxford (where he himself went) and the insinuation that Oxford or Cambridge alone would of course have made even Shakespeare that much better.
At least he doesn't go off the deep end and suggest that without this education Shakespeare couldn't have existed.  If anything this seems to me like a positive quote, doesn't it?  Here you've got an Oxford-educated man who willingly admits that the "mystery" fascinates him, and he wants answers.  He's spent his life with the belief that only those with elite education can change the world, and yet Shakespeare is the obvious exception to that rule, and Dawkins wants answers.
I suppose if you're in a grumpy cynical mood you can read between the lines and argue that this is indeed an authorship attack, and that the sentence following the quote above could easily be, "Perhaps you weren't a humble country boy after all, perhaps you did study at Oxford..."
What do you think, does Dawkins go into the Denier camp, or is he just being attacked for daring to approach the question?