Seen this theory a lot today. The problem with the devil fruit part is that in the series devil fruit maintains the japanese which is akuma no mi or something close to that, just offering a counterpoint. It could end up being true. Also saw an interesting one that D is associated with Davy Jones, evil spirits of the sea, and Davy Jones Locker. We saw early on that in OP there are Davy back fights, which originated on Pirate island.
Anyway the story we're hearing from Corazon is what was told to him as a child by the Celestial Dragons, who refer to themselves as Gods. In that case the D's are enemies to the Celestial Dragons who we know are actually human. It's probably that the D's were the former rulers or protectors of the ancient kingdom before whatever occurred during the Void Century. Which would make D's the natural enemies of the "Gods" which in this case is the Celestial Dragons, humans claiming to be God. I think D deals with the name of the kingdom that was destroyed and is now covered up and denotes those who descend from the original rulers of the world. Partially I think this because Clover was killed right when he was about to say the name of the Kingdom, also it's said D can't be fully understood without reading the Poneglyphs, and last D has also been mentioned with some type of inherited will, that has given all D's except Blackbeard similarities. I like to think it's as Marco mentioned it's Blackbeard's irregular body that allows him to eat multiple fruits. Plus racking up devil fruits to give Luffy a power boost wouldn't be cool to me.
Also I like to think Roger rose to be Pirate King without the aid of a Devil Fruit since it hasn't been mentioned that he had one. Also Garp is devil fruitless and one of the strongest characters in the series.
Who knows though you could be spot on with your theory. That's what makes One Piece fun, it's unpredictable and has plenty of mysteries. Just thought I'd offer an alternative theory.