For ATB the cooldown is visual and gates your turn, which gates your skills.
In the end this is all really just semantics, but again ATB (as implemented in most FF games) is best described as a turn cooldown, not a skill cooldown. In many of the games it is just a flexible, visual way to manage turn orders without relying on rounds -- where you simply rotate through each character. Some FF games freeze time once the ATB is full; others provide time freezing as an option (similar idea to previous Baldur's Gate games having real time + pause layered over a traditional round-based system).
Because ATB doesn't have rounds, it's possible for one character to have 2 turns (or any other arbirtrary number of turns) in the same amount of time another character has 1 turn. It's more realistic and provides a clearer advantage for fast characters, but it's more chaotic and can be harder to plan around. Either way, these are still "turns" in the tradional sense, and it's the turns that are experiencing a cooldown.
Yeah, I referenced the games to have a reference where cooldowns are more frequently used. The intention behind it was to be more like real-time with pause. Here's how ATB was described back in the day. (Page 16)
SNES Manual - FFIV (Start of ATB)The idea was that the battles would have a higher sense of urgency. I totally agree that it gives an edge to faster characters and could make planning ahead more difficult.
Anyway, not a system meant for BG3.

Also to add to that and getting on-topic. BG3 has a really complex system adopted from DnD 5e. DnD expects the players to plan ahead and is very much balanced around that. Changing spells to be gated by cooldowns instead of long rests would disrupt this. (Just as short rests recharging spells like long rests currently does).
TBH long and short rests will probably need to be changed again, I think by and large we can agree something is off about it. Too often we're not planning ahead and repeating the same mechanics like a chore.
Whether the game should be changed to be truer to DnD or changed for the modern video game experience, I don't know yet. The changes that would benefit the game are those that will help the player plan ahead and those that would inspire the player to repeat tasks again.
For example,
Change short rests to only restore health. (Keep at 2)
Change food to never restore health.
Change food to at first have no effect, but when made into a dish can restore 1 spell slots.
(different dishes will replenish different levels)
Add in: Characters will not want to eat again until after a short or long rest. (after eating, character gains a "full" status, after resting loses full status)
Change food to only be consumed outside of combat.
Change long rest to only occur in safe zones/rest points.
Significantly drop the number of spell scrolls that appear in the game.
These changes would improve the game, but I'm not sure if they would fix the issue. (I think they would help the player plan ahead, but the mechanics could still feel like a chore). I'm sure there are other aspects of the game that will need to be changed as well, and restoring 1 spell slot might be too few for the modern gaming experience. The example is mostly food-for-thought.