Wait a minute...is this a plot-hole (Austerities of Wind and Main Scenario-related)?

[Exodus-E]

What we confront is the essence of Garuda we got from her defeat, even though she was trapped in Ultima we got the crystal of wind directly from her body so that should do the trick.

Really?
I thought the austerities worked without the need of an aspected crystal (in fact, I don't think the Summoner quests ever mentioned it); instead, simply needing the defeat of a given Primal, and call upon their essence from an area where their aspect is strong.

Unless I'm missing something else (because, in Garuda's case, if her whole essence was inside Ultima Weapon, the we shouldn't have been able to call its Egi...would we?)...

While I can't think of all of the details the short answer is, yes some of her essence remains for you to do the trial.
Basically as a summoner that is what you are doing harnessing a small part of the primals essence to use.
And while I am not sure how far into the story you are, like fighting Mog or Shiva, you learn that the Primals aren't what we initially thought they were.
While I cannot go into too much detail, as the lore is still debatable but for the most part accepted.
The primals aren't actually entities that are actually summoned but ideas and beliefs brought to life by aether.
So while Garuda was "absorbed" she still exist and could be summoned again.
But the best reason for why she wasnt, immediately, summoned again is probably because watching their "god" be defeated before their eyes, broken the ixal's spirit.
They could summon her again, but they briefly lost the desire to.
Sorta like when you lose the will to fight during a game or the will to study for a test, similar principle, you still can but the will to do it is gone for a bit.
I hope that helps a little, if it doesn't confuse you more, lol.

That could be it.
I know that the lore behind them wasn't fully revealed/expanded upon (beyond "entities made manifest through prayer and aether"), but if we do have some of the Primals' essence still available out there (and/or within us), that may explain how we were able to call forth an Egi from Garuda, despite her being trapped at the time.