It doesn't make sense (to me).
The Time/Space/Star Mage part, that makes perfect sense. It meshes very nicely with Sharlayan as it has thus far been presented to us (a nation of scholars, academics, scientists, and mages). That part is all very nicely quantifiable, observable, and logical (within the setting).
Its the Fate/Luck/Future-Seeing part that I cannot wrap my head around. At least, I can't wrap my head around this, because this class as we know it and play it comes from SHARLAYAN, a nation of Scholars, Academics, Scientists, and Mages, people who would outright dismiss fortune-telling as self-flattering nonsense by attention-craving mages.
The Lorebook tells us that Lewphon conducted a study of the ancient civilizations and their prophesying and (somehow) found conclusive evidence that there is measurable, quantifiable data to indicate it is not nonsense, but accurate in some form, at least when applied to the stars.
This is where my issue starts cropping up.
This ability to foresee events to come will have one of two primary outcomes of perspective for the players. It'll either damage the integrity of the existing lore and set a terrible precedent for all future events by establishing this as an infallible source of meta-knowledge, or this ability to foresee events is not accurate and thus not significant in the slightest, despite being presented to us in the questing for Astrologian as highly significant.
Let's look at some established lore events.
The Exodus of Colonial Sharlayan: If the Sharlayans have the ability to foresee the future, then the decision to abandon the colony was undoubtedly a result of seeing the encroaching doom of Eorzea.
Viewing this through the first perspective utterly destroys the integrity of the character of Louisoix. Prior to the Astrologian Class, Louisoix was presented to us as an idealistic, benevolent figure who strove to avert a massive catastrophe, but failed. However, a man of his years, of his experience, and from Sharlayan, would have to know, under this perspective, that the Astrologians are NEVER WRONG. So, knowing this, he chose to put the lives and careers of those nearest and dearest to him in jeopardy for... what? He knows he can't change what is to come. The Astrologians saw the Calamity, he can't stop it. Why try? He's a Sharlayan, an academic, a scholar, a scientist. 1+1=2. You can't make it add up to 3 no matter how much you WANT it to, or TRY to make it.
Viewing this through the second perspective, the Astrologians foresaw the destruction of Eorzea and/or Garlemald conquering Eorzea. Except... neither happened. So... why did they abandon the colony? Because they chose to put their faith in Astrologians? But if this is the standard of Astrologian Foresight, that it is fallible, why put any stock into it at all? Why would it be just a significant aspect of the job's lore?
The Isle of Val: As anyone who has completed the MSQ up to 2.5 knows, the Isle of Val, which was the home of the Students of Baldesion, a Sharlayan institution, was wiped off the face of the world with only -1- survivor.
Looking at this through the first perspective, we must ask why this was allowed to happen. The Astrologians would KNOW Val was doomed, along with everyone on it. They spared the Colony the Seventh Calamity, but decided to cut loose the Students of Baldesion? Did the Students of Baldesion have not even -1- Astrologian among their ranks? Did they decide to IGNORE what was going to happen?
Looking at this through the second perspective, we have to ask if the Astrologians even saw it in the stars to begin with. If they didn't, then why is this fortune-telling aspect of Astromancy so emphasized?
Final Thoughts: As far as the story goes, I do believe there is a place for a Fortune-Teller Class that attempts to divine the future, but I cannot validate it as part of a nation like Sharlayan, which should, "Know Better." The only way this can be validated as presented is if its some kind of fool-proof, 100% accurate art. As we see, that damages the integrity of the existing lore, and will call into question everything that happens in the future. If this process of seeing the future in the stars is NOT accurate, however, then why would the Sharlayans give any tangible measure of accreditation to it? It should be an idle curiosity at best, a topic of scorn at worst.
Why is this part of a Class from a nation of Scholars, Academics, Scientists, and Mages? Being able to, 'See the Future,' is a difficult enough variable to add to a narrative without putting it into a culture that would reject the notion.