Chocobo Paradise / Fat chocobo as your job quest giver, I am a fan of that lol. It'd be fairly unique, I think, that a job quest and beast tribe are intertwined and related content.
I think that could make managing chocobo abilities on the mount bar really easy, but a few questions too~!
By the mount bar I mean we all know jobs have that "choose your abilities from other jobs that you want to use" section and we all select all 10 available because.. why not. Here however you'd have a list of abilities you've earned (over time very likely over 10 options) and you then select the skills and what have you that you want on the mount bar.
I guess the question would come in, since you said all the above content - are you including the mini game and any combat that may relate to that? I can imagine once you begin pecking about the bar switches to hot and cold related skills (if SE makes it that interactive), which would be easy to manage but also doesn't really require you be on your chocobo job. Are you envisioning the breeder / racer using chocobo theme'd combat abilities? I can definitely see the chocobo getting a huge power boost when you're in this limited job state, that and being able to stay out as long as you want them to. In fact I could see if you were in your chocobo limited job that you could even have multiple chocobos out at the same time! That'd be pretty interesting, wandering around the world with two-three of your own raised chocobos (perhaps finally a way to solo those FATEs that aren't normally solo-able lol). What sort of gear would they wear though? Perhaps DoL? Higher perception increasing their abilities (that care about stats), and having a chocobo out could be another stat booster?
Serious tangent here but if there are combat abilities one of the "ultimate" abilities for your limited job should be being able to transform into a chocobo for a short time lol (like 10 min cooldown, lasts 2 minutes - or even more exaggerated). Giving some nice abilities and a huge passive movement speed boost (now you look like a pack of chocobos running across the lands, which imo would look pretty cool in game).
Certainly changes some things when the content is bound to a job but it could also quite nice. I think one of the biggest QoL I can think of immediately from this would be instead of having multiple chocobos you have to rank up one at a time, you can go out with multiple and level them all together (besides the improved training that chocobo paradise offers as well, mentioned in OP). And the biggest stress would be if you had to switch to the job every time you wanted to do some basic chocograph, because extra buttons and it might mean you can't do it due level (say for example the job is only level 50 and you're in a level 80 area). Although on that point of stress I can imagine that the open world version doesn't need to change your job, just switch the ability bar and if they want to switch you to your job it could be when entering special forests/content areas (like for job quests it's a shimmering light that asks you to switch to the proper job and says "are you sure you want to enter?").
One of the unfortunate side effects would be you wouldn't get exp on your job that you use in other content, so I know this is talking about rewards in a bit too much detail for this early of a concept but I'd like to see some sort of choco sphere as a possible reward for some of the content that boosts exp and can stack with other exp boosting items then (to help offset that exp shift away from your other jobs).
The other concern being we have racing already in the golden saucer, I could see the golden saucer being a way to exp up your racer but I'm not sure it would be good balance to allow your job to affect the chocobo race in the saucer (although a lot of times you race against npcs, in which case whatever lol). I however wouldn't be opposed to SE remixing all chocobo content into a neatly organized and packaged system, that including Golden Saucer stuff would be okay too.