I feel like they did it just to give us a way to defeat Corypheus. I mean... it doesn't make a whole lot of sense in-game, does it? He can't be killed like an archdemon, but he can be killed because he was... uh, sharing his immortality or something with a dragon? Something which has never been brought up as possible, much less desirable, before or since? Okay...
Really the whole plot with Cory was kind of a wreck after Haven. Makes me wonder if the writers were so focused on Solas that they barely gave any thought to his set-up villain, or if they originally had something else planned and had to slap together the Cory plot last minute.
No, he wasn't sharing his immortality with a dragon - he was immortal long before that. That's the reason why he was locked in that Warden prison.
Morrigan explains that it's his pride that will doom him - he got himself a dragon and invested part of his power in it NOT because it's a horcrux, but in order to emulate the Old Gods (or so he thought, because it's suggested that he learned the knowledge of how to do it through the orb). The dragon was very much an expensive "status pet".
Thing is, that in order to create such pet, he had to put some power and his own being in it - and when we kill that dragon, part of his soul rushes back into him, temporarily disrupting his ability to body hop.
It suggests two things - that his, or Archdemon's immortality, is effectively similar to that of ancient elves/Evanuris (how else the well Of Sorrows would have information on how to beat him? And does that mean that Blight magic is a corrupted form of ancient elfy magic?), and that Evanuris either had such pets themselves, or it was part of something more elaborate. It may even be that this is how Mythal was killed - they killed a powerful pet of hers and then struck the killing blow.
In other words, rather than being underwritten (that part at least), I think the way Corypheus uses the power (he barely understands) and how it ends him reveals hints about bigger plot.