Even if there was a timer, I don't think I would mind as much as I did in XI.
Firstly, we have chocobo breeding (meaning we can have more than 1 chocobo), unlike our companion where we only have one.
This means that even if there was a cooldown timer, after a while we will be able to switch between chocobos and moreso switch chocobos depending on what we're doing.

Some examples:
If you are a gatherer: you can raise multiple storage chocobos, when one is full you switch to the next.
If you are solo fighting: you can raise a healer for when you do melee and when that runs out you might have a tank choco raised and you can switch to a support class or mage.

TLDR: We can raise multiple chocobos. The cooldown won't be as bad once you have 2+ chocobos.

This is all of course assuming that you can summon any of the chocos you have and not just one.

My fear is that it'll take a full month of adamant tapping enter before finding out a have a plain old yellow chocobo instead of the damned red one that I NEVER get. EVER