I honestly don't think the the in question is gratuitous, and not even superfluous. And the first word, Oh, is one syllable and is sung with two pitches anyway, not one. Still, your theory is a good one, even throwing "Oh" out of the argument, especially if the first versions never contained the the in question. So far, every version I've found online includes that word that I'm fast becoming sick of referring to!

Best regards,
WW

I would be interested in seeing a version in which the the in question had been omitted or hadn't appeared in the first place, but, based on what I read today, it sounds as though several versions circulated all springing forth from the head of Mr. Key.