You're misunderstanding that quote - or at least we are both interpreting it very, very differently.
For reference, Mide and Dayan's entire conversation from Judgement Day:
MIDE
My love...! What...what is this place?
DAYAN
How best to put it? We stand inside a mathematical simulation, calculated and projected by the device at the heart of the colossus. One might equally say it is the dream that Alexander dreams...
MIDE
And what of our companions? Are they...?
DAYAN
...Their fate was not mine to change, Mide. All that came to pass did so for a reason. History is as it was─as it should be─free of the paradoxes that spelled its undoing.
From this place─unfettered by the mortal construct of time─Alexander looks out upon past, present, and future, seeing infinite possibilities. I see what it sees, and feel what it feels─this perfect machine, born from yearnings for an ideal world.
Oh, if you could see the worlds we have seen! A world in which the Illuminati rule history with an iron fist, every nation brought under their yoke. A world in which Alexander spread wide the wings of time and swept the lesser moon from the heavens, averting the Calamity...
Alexander dreamed all the realities imaginable─all the realities mathematically computable─and in the end, reached a single, logical conclusion. It would change nothing, and erase itself from existence.
MIDE
But...why?
DAYAN
Alexander possesses the power to travel through time and space, and reshape history for the better─but such power comes at great cost. The sheer quantity of aether consumed in the process means that Alexander itself would─mayhap not immediately, but inevitably─bring ruin to this world. This perfect machine, this supreme manifestation of logic and science, deemed its own existence a threat.
And so it chose to do nothing. To leave history untouched, and the future in the hands of man, with all his imperfections. Such was Alexander's divine judgment.
A time will come when the fate of this world is placed in the hands of one warrior. For reasons hidden to me, the future from that day forth remains shrouded in mystery─beyond even the colossus's ability to calculate. And yet Alexander chose to believe in that woman, and the light within her.
MIDE
I met that woman, my love... I believe in her, too.
DAYAN
There was but a single time Alexander was spurred to action─not to change history, but to preserve it. The summoning of the colossus, and the events that followed, had potentially disastrous consequence for our reality. Its fabric strained to accommodate an infinite number of potential futures separated by nary a thread.
Were the wings of time to fall into the hands of the Illuminati, the repercussions would be dire indeed. History would be rewritten over and over again, each time bleeding the land of aether. And in the end, the colossus would usher in another calamity.
To prevent this tragedy─to preserve the circle of time as it had already been set in motion─Alexander sent forth a humble servant to do its bidding.
A clockwork coeurl, an eternal child, to gently nudge history back onto its proper path.
And now but one task remains.
Having become one with the horn, the core is well-nigh indestructible. It would only regrow itself until it bled the world dry. And so we must heed Alexander's wish, and seal it away─far from the reach of man.
A closed pocket of space-time, a fraction of an instant that will repeat in perpetuity for all eternity. That is where the core must go─and us with it.
MIDE
Don't you see, my love? We are already there. In coming here─to you─I have closed the circle of time. I am all but certain of it.
When I accepted the horn all those years ago, the man who gave it to me said that it had the power to grant any wish. And so, as I took it in hand, I prayed and prayed with every onze of my will for the one true desire in my heart...
To be with you...forever.
Alexander's "dream" is not a dream at all but a computer simulation explained in poetic terms - alongside the factual one.
There was no alternate timeline that had to be erased - those "worlds" never truly existed, but were merely simulated. Alexander calculated all the possibilities of what might happen.
It calculated a possibility that the Illuminati would win and enact their plan.
It calculated a separate possibilty that it could interfere in the timeline and avert the Calamity.
But in the end it concluded that it should not change anything at all.
The "single time" that Alexander acted is a little vague, but was either our brief trip through time, or (more likely) deliberately sending Shanoa/Schrödinger out to keep the time loop "on track".
They actually do mention that Alexander's continued existence would risk another calamity - but not because of time travel itself. Instead it's the amount of aether that it would drain from the land in doing so.
Also worth noting: the fates of Mide and Dayan's companions "were not his to change" - ie. no meddling in pre-established events, even if he wanted to save them - and he seems to regard history as currently "free of the paradoxes that spelled its undoing" even with the time loop in place, which suggests that the mere existence of a time loop is acceptable, as long as it plays out neatly and doesn't break anything.
And actually, Mide's closing line suggests that the entire loop wasn't a thing that happened inexplicably, but was orchestrated "from the middle" by Alexander itself.
This isn't the source of that aether, and it isn't covered in a single "off-handed mention" - it was discussed in more detail earlier in the story. Basically there's a magic artifact in the core that is drawing in large amounts of aether, causing it to grow rapidly, no time travel required.
In The Coeurl and the Colossus, Y'shtola questions Mide for more detail about the summoning, and she explains:
MIDE
I fear it will not avail us, but...very well. It all began with a curious encounter in my younger days. I was out playing on my own one day when a stranger appeared before me. He told me anything my heart wished for could be mine if I but heeded his words.
In my youth, I did not fully comprehend what the man was proposing at the time. Nevertheless, his words seemed to burn themselves into my very soul. “Reunite the scattered shards of the Enigma Codex,” he said, “and perform the sacred rite before the ruins that yet stand strong against the river's flow. Then, and only then, shall your hopes and dreams become reality.”
With that, he handed me a relic─a horn, with a tip of metal─which seemed so foreign as to hail from another world. It would serve as a catalyst for the “summoning,” he explained. “A gift from your friend Travanchet.” And then he was gone.
Y'SHTOLA
Travanchet? I know the man. Our paths crossed during my time in Limsa, in the days before the Calamity. I might have guessed that the Ascians had a hand in this.
The horn of which you speak was once rumored to reside on Seal Rock. It is a lost relic of the ancients with the power to manipulate the aetheric energy in the very air around us. The Scions have long pursued the horn, fearing the devastation that might ensue were an artifact of such power to fall into the wrong hands.
Finally, I do begin to see...
If I have the right of it, Alexander's third core is using the horn to draw aether from far and wide. 'Tis little wonder our plan failed. Even without moving an ilm, the primal possesses the means to drain not just this land of aether, but our very star.
...oh, and on a totally random note, from checking the script, the cat's name "Shanoa" finally makes sense. It's tagged as Chat Noir.
I'm not sure how I'm stunting future stories, and I'm not necessarily proposing "controlled" time travel either, even if I've maybe talked about it as an example of time travel before. (My example about Gaius was just, well, an example of how time travel could be involved in pre-existing events in a story. I'm not proposing that's what happened.)
I think I keep wandering between my actual thoughts on the plot and my general thoughts on writing timetravel stories because you keep proposing ideas that aren't what I'm thinking of for this plot but I do have thoughts on them regardless. And then I go off the track.
Anyway, for consistent time travel to a functional game zone, I would expect a fixed portal that moves you from "Current Time Bubble A" to "Future Time Bubble B", and the pre-established handling of time within the game continues.
What it would allow is a reversible "timeskip" to a future part of the timeline.
Then the overall timeline would look something like:
(( [1.0]---[ARR]---[HW]---[SB]---[5.0 TIME A]---[6.0]---[7.0]---[etc.] ))------???-------(( [5.0 TIME B]---[6.0 time plot] ))
Future stories in the main timeline just keep happening in the 'red bubble', always being placed prior to the first event in the 'blue bubble'. They'll never join up or overlap.
Ultimately it probably doesn't make much difference to the function of the game. It would act like another zone, just with different storytelling rules. You could even make a lore basis for aetherytes and teleportation working normally between past and future zones by saying they can be mapped across space-time rather than just physical space - it's just that nobody ever timetraveled before to discover they work like that.
Remember that I am coming up with this from the starting point of "the story seemed to be hinting at time travel - how would they make that function?"