The Ishgard restoration FATEs were not terribly exciting even if you did manage to catch one:
Interact with items at A.
Move to B and interact with another item.
Go to C and interact with yet another item.
Rinse and repeat until the FATE is finished.
Get a few seconds worth of cutscene as another structure is completed - which you can see in its completed state anytime you visit the Firmament.
Profit!

(There almost certainly were minor variations between the different FATEs, but I doubt there were major ones.)

If you want to see how work is progressing you need only visit the Firmament at regular intervals to see how it changes over time. If you want to help restore it, the real work consists of turning in all the collectibles.


Someone starting play a year or two from now will probably not be able to see how the restoration progressed, just as players starting out now won't be able to see how Mor Dhona or Idyllshire changed over time.