Haste in FFXIV doesn't work that way, though. Haste increases the number of actions you can do by X%, it doesn't reduce delays by X%. See: Monks and Ninjas with their haste buff and Arrow back in the day. It didn't break the game.
The real problem with castable haste buffs on other players is that this game has some rigid timings for jobs and an outside haste can royally mess with it. For example, if your rotation requires you to be at a certain GCD when X ability comes off cooldown, but instead you're 1-2 GCDs ahead because of a haste buff, it can really screw with you. In addition, Haste buffs can vary wildly in how effective they are. Some classes can capitalize off it, others... really can not. See: Skill and Spellspeed, and how most jobs avoid stacking it past a certain point due to it just not helping as much.
Could it be worked in? Sure. Would it require a massive overhaul to every job in the game and how oGCDs, floored speed GCDs, and cooldowns work? Yes and... euuuuuugh.