Yay essaying:
I have been hoping for updates to the Chocobo Companion for a number of years, sadly I think it is one of those things that has been relegated to the classification of a "finished system". While not as completely fleshed out as it could be, it already accomplishes the job it needs to: Assisting the player when they're alone or in a small party in the overworld, particularly in tackling FATEs. This being the case, there's little incentive to improve or change it as the invested resources wont have much player impact on the gameplay loop that it's used for.
I feel for the chocobo companion to be touched, there needs to be a greater need for the companion to exist beyond helping with the occasional FATE. There have been calls since ARR to allow us to take the chocobo into dungeons, but these days the question is "To what end?". It made more sense in earlier expansions given the more sprawling nature of the dungeons and the more "rooted in reality" locales. But these days? Taking my chocobo into something like the Dead Ends? It's difficult to see.
Now, if they wanted to create a string of side content specifically designed to be done with you and your chocobo, which would require an overhaul of its systems, that I could see happening. In fact, I would love to see a deep-dungeon-like mode where just you and your chocobo, in a solo venture, journey ever deeper to uncover secrets and face trials together. Perhaps to find the true Fat Chocobo as you do in so many other Final Fantasys. And as a reward, your chocobo is perhaps a little stronger or has sharper looking skills, or gains small levels of greater customization (such as multicolor options like dying the tips of its wings different colors, giving it sparkles, or access to special colors). This would put it in the space of an achievement flex for players. Like those that have earned a special mount or those that have such coveted titles as "The Necromancer".
Now these things could be added to the system as is and be rewards for further level increases, but I think we would find that simply leveling your chocobo as you always have is not particularly engaging or meaningful on its own. A similar problem has stumped the devs for multiple expansions when it comes to increasing our status within the Grand Companies. We can currently access the rank of "Captain" but the 8 ranks above it, time and again the devs have said they want any further escalation of ranks "to MEAN something" and they're not sure how to best accomplish that yet (or more likely, do not have the resources to pour into what is more or less a relict system that not very prevalent to the current player experience.)
That said, I think a special set of dungeons, an adventure, or even a deep dungeon that you take on with your chocobo would be really cool and I hope someday something more comes of the system. Fingers crossed.