Chocobo companion timer.

[Enkidoh]

If I had to hazard a guess, I would say it's deliberate to actually disuade you from having your chocobo out at all times - they don't want you just spamming open-world content over and over again for hours on end with your chocobo, but actually varying what you do (e.g do a few FATEs, then move onto a dungeon or two, then maybe work on some sidequests or something similar) - which is why you cannot take your chocobo into instanced content. Your chocobo companion is simply meant to be a help in certain open-world situations (such as open-world solo fights or FATEs), but not as a crutch to over-rely on.

Really, it's not that hard to simply buy a stack of gysahl greens from the many vendors, then map it to your pet hotbar and simply hit the icon when the chocobo's timer gets low, which again is probably what SE were thinking when they designed the system. It doesn't need to be even more simplified than it already is.