Once upon a time, FF14 didn't have enrage timers, at least not in the same way we have now. When 2.0 was released, the end game consisted of the 3 hard-mode primals (Ifrit, Garuda, Titan) and Coils 1-5. Of those encounters, only Twintania had what we would call a "hard" enrage like is commonly seen nowadays. Everything else either had no enrage (the primals, Coil 4) or a soft enrage (Coils 1 and 2), and in the case of Coil 2, this led to the very real problem of dropping a dps for another healer to ensure a nice, safe, guaranteed clear. This was seen most commonly with the Coil 2 enraged strat, whereby you'd wait for the timer to tick down before engaging the final node and just utilize a combination of tri-healer with BRD MP refresh (via Ballad) to out heal the incoming aoe damage while also avoiding Allagan Rot.
SE doesn't want that. All of the design decisions that exist in encounters today are done with the idea of enforcing the 2/2/4 tank/heal/dps composition. Tank busters and swap mechanics are there so that you need to bring tanks (i.e. to avoid stuff like solo tank Zurvan/Ravana as was seen a lot in HW). Healers are often the designated jobs to be targeted for a number of key mechanics throughout a fight, many designed specifically to make the life of a healer more difficult (Seiryu's split aoe stack mechanics in the final phase making it impossible to solo heal). Dps are often given their own designated mechanics as well nowadays, so that the measure of a good dps isn't simply how much damage they output, but their ability to do that damage while also juggling the mechanics the fight imposes upon them. But unlike with a tank buster or a big heal check, those mechanics don't *require* a dps to do them. The only thing that can force the requirement of a dps is the one thing that all dps bring over tanks and healers; more damage.
Lamenting the state of the game in regards to enrage timers will do nothing. They are here to stay, simply because if they weren't then we would run the risk of the playerbase yet again excluding jobs in favor a safer, more certain group comp.