TotK's past happens relatively shortly after the foundation of the Kingdom of Hyrule. Hyrule Historia is very clear that Hyrule was founded after the Era of Chaos, before Minish Cap. Note that the foundation of the Kingdom of Hyrule is a different event than the people of Skyloft colonizing the surface and calling it "Hyrule" (again, that's from Historia), which happens at the end of SS.
This placement also explains why nobody mentions the Triforce or the Master Sword: to end the Era of Chaos (when the Interloper War from TP happened), OoT!Rauru hid them away. They aren't featured in Minish Cap or Minish Cap's backstory (the Hero of Men) either. MC!Link and the Hero of Men resorted to the Picori / Four Sword instead. Thanks to TotK, MC fits better in the timeline. Before, it was sandwiched between SS and OoT (well, there is FS too, but that game barely had a plot), two games that featured the Triforce and the Master Sword prominently. MC felt like a spin-off given a place in the timeline. With TotK, we have another game taking place roughly around the same time (very early days of Hyrule) in which the people of Hyrule are unaware of both of them.
TotK!Rauru and OoT!Rauru are two different characters sharing the same name. Just like OoT!Kaepora Gaebora (who's OoT!Rauru in disguise) and FSA!Kaepora Gaebora (who's just a random talking owl). They are both sages of Light, so perhaps "Rauru" means "Light" in Ancient Hylian? I mean, we had two completely unrelated races named "Zora"; we can have two Sages of Light named "Rauru".
What about Ganondorf? This is what's throwing people off. We are so used to OoT!Ganondorf as the character's first incarnation, it's hard to think that might be false. And TotK shows it's false. TotK!Ganondorf is Ganondorf I, OoT!Ganondorf is Ganondorf II and FSA!Ganondorf is Ganondorf III. This also explains Twinrova's plan, they want to revive the "spirit" of the original 'Dorf. So, wait, does that mean have Ganondorf I's corpse buried deep below Hyrule castle during basically every game in the series? Yes, just like we had the sleeping Zelda during Zelda II (and, to a lesser extent, the wandering spirit of the Hero of Time between MM and TP). AFAIK nobody in-game draws a connection between Ganondorf I and Calamity Ganon. We also know that Calamity Ganon is the result of countless battles between Chosen Hero of a given era and Ganon, who, logically, cannot be Ganondorf I, as he's stuck below the castle. Both Creating a Champion and BotW itself (via the Zora tables) suggest Calamity Ganon is OoT!Ganondorf.
Yeah, Ganondorf I and Ganondorf II's plans are awfully similar but, look, Tingle also behaves the same in every game he appears in. This also means that TotK!Imprisoning War and the post-OoT pre-ALTTP Imprisoning War are two different, similar events sharing the same name. This is nothing incredible if you think about it. Our world had two World Wars, and both of them can be described as "US+UK+France+Russia against Germany; Germany invades Poland and Belgium; US enters late but plays a vital role".
So, tl;dr:
• A game showing what happened shortly after the foundation of Hyrule takes place when the official timeline puts the foundation of Hyrule: before MC, after the Era of Chaos;
• Nobody mentions the Triforce nor the Master Sword because Rauru hid them to end the Era of Chaos;
• The only other game where this happens? Minish Cap, which also takes place in the early days of Hyrule;
• As long as we accept the TotK incarnations of Ganondorf and Rauru as separate characters from their OoT incarnation, there are no plot holes, and everything fits together nicely, giving further context to Twinrova and explaining where Calamity Ganon comes from.