"why is the Temple of Time missing during TMC and FS" is a question I could anwser.
Why should TMC and FS take place after the construction of the Temple of Time ?
Why not before ?
As FS is taking place in FSA as the prologue confirms while also taking place in the Downfall from the existence of the realm of memories in the anniversary edition, containing only games from the Downfall.
So ... Is FS what HH pretends OOT is ? The starting point of the split ?
ALBW kinda disconfirmed HH which was saying that Hyrule was established after the wars for the Triforce and its sealing to the Sacred Realm, as ALBW mentions that a King ordoned himself that the Sages shall seal the Triforce.
If there is a King, then there is a Kingdom, and the castle was depicted in the paintings. So the Wars of the Triforce took place after the establishement of Hyrule.
I strongly believe that the Force Era, TMC and FS, took place in this few centuries young Kingdom of Hyrule. The "Hero of Men" ? The Hylians thought of themself as mankind, unaware of the human tribes beyond their land.
Fi was taking care of the remains of Demise thus making the Master Sword unavailable. Making the Picori Sword's existence quite important in a time where nothing else was strong enough in order to defeat the demon horde.
Thus, the wars for the Triforce took place after and with it the split between the DT branch and the AT, CT one.
The DT one lacking the Temple of Time.