I would have to say that the hardest part of leveling your Chocobo is that you can not do it while waiting for a Duty.
Why can't the Chocobo be written in as a Summoned pet like the Arcanist ?
Hard enough to earn the Chocobo experience as it is but you can't have him out while awaiting the duty ... which depending on your role as a DPS can be a half an hour or more at times awaiting specific dungeons.

Very Respectfully
Me