The Dragon as the name of a trope is an ancient relic dating to the earliest past of TV Tropes, a time when trope names didn't need to make sense. It's a terrible name that people only remember because of its common occurrence and egregious lack of sense. I happen to know that the trope was originally named by some random person after a work of fiction where the Dragon was a dragon. The other explanations are Retcons. In particular, drawing from Campbell is a post-hoc justification, and doesn't refer to the same trope anyway as far as I can determine.
I looked throught the examples, and it looks like only about 1% -- under 2% at most -- of the listed Dragons are actual dragons. Some of these mentions are misuses of the trope, as people scramble for find dragons that could through some twist of logic actually be called Dragons (and not just because Tolkien liked to capitalize the word). There are also many mentions about things like how the work has a dragon who nevertheless isn't The Dragon or how The Dragon isn't a dragon but instead has a sword called The Dragon. There are also several examples of a situation where a non-Dragon dragon is the Big Bad and has a non-dragon Dragon.
I know fixing all this widespread nonsense is a huge effort, but I think it should be done. It's really not too different from replacing the old, non-standard titles for trope listings and putting those tropelist templates everywhere. I have a lot of free time at the moment and can work during the hours when the site doesn't tend to choke on gateway timeouts.
With The Dragon there is also the issue that two related things are conflated: the Top Enforcer and the Special Pet of Doom, as they could be called. It's the difference between Sauron and Gaurhoth in Tolkien's Beren and LĂșthien in The Silmarillion. Sauron is a mighty sorcerer in command of his own fortress that guards one approach to Angband. Gaurhoth is a terrifying super wolf, whom Morgoth raised personally and later chained to guard the gates of Angband, but Gaurhoth still doesn't show itself to be more intelligent than an animal and is in charge of nothing. I would also call the literal dragon Wong Bongerok from The Fortress Unvanquishable, Save for Sacnoth an example of the Special Pet of Doom. Even though Wong Bongerok is intelligent, its master still treats it as a pet and Wong Bongerok is fine with that, apparently because of its non-human mindset. In video games there is also a difference between how you may meet the Big Bad's Top Enforcer in the game, doing plot-relevant things like ambushing you and appearing as a Climax Boss in connection to some big revelation, while the Special Pet of Doom would just be some non-human (or non-Elven, etc.) beastie that you need to fight through just before you get to the final boss.
I think splitting the trope though is not nearly as much a priority as changing The Dragon to something else, which I think should be Top Enforcer, given how that seems to be by far the more common form.
The same sort of name change would of course go for the other tropes with names based on The Dragon, so that for example Co-Dragons would become Co-Enforcers or something. Noble Top Enforcer at least already exists, so that one won't need any changing.