You also forget Celestial Opposition, which is about as close as it'll get in the game to any Stop spell in terms of gameplay mechanics (boss doesn't move or attack, buffs get extended, so it's as if time from the enemy perspective looks like time got stalled a few seconds).
The loose consideration that is used for the "it's a Time Mage" argument is that Time Mage is effectively an Astrologian on steroids, much like how any sort of necromancer is effectively a Black Mage on steroids. It's also understood that Time Mage as a standalone concept of just buffs wouldn't work in this game due to the necessity of it being forced into a trinity role. The question then is whether or not Time Mage can be implemented with enough of its identity present while still appearing distinguished from its star-crossed counterpart. Shockingly enough, this wouldn't be the first case where the two jobs were in the same game, as Bravely Second also presented both Astrologian and Time Mage in the same game with completely separate identities.
Could that be done in this game? Definitely. Much like how Black and White Mages are presented in lore as two sides of the same coin, Astrologians and Time Mages could be presented in a similar fashion, perhaps with an added flair of school rivalry in one of their stories. A better real-world analogy would be to compare Astrology to geometry and Time Magic to calculus: two very different paths, perhaps one more difficult than the other, yet still leading to the same intellectual destination. I even made sure that would be consistently displayed when I made my concept for Time Mage.
Does the concept proposed by the OP provide that required degree of separation? Although I'm not quite sure of some minor elements of the concept such as why it requires melee and the use of older advanced spell terminology (i.e. Hastega rather than Haste II for this game), this concept still does present itself as a viable case that Time Mage could exist in FFXIV even with the presence of Astrologian and the handful of iconic elements it has "borrowed" from the job.