I have a lore related question then:
Phoenix Down item description: The billowy plumage of the fledgling Phoenix─a legendary bird that, upon each fiery death, is said to rise from its own ashes in a literal “blaze of glory.” When placed upon the body of one who is incapacitated, it will instantly revive the person. While some cultures insist that the tail feather's restorative properties trump those of down, these claims have yet to be proven in any sort of scientific capacity. ※Cannot be used during battle.
So, are these parts of our beloved Archon-as-Primal, akin to the Heart of Titan, Garuda's feather etc that are left behind on other Primal's defeat? Have there been other Phoenixes present in Eorzea? There must have been enough parts of a Phoenix to be studied, at least...
Also, it seems people do believe, or know of Phoenix, at least: "a legendary bird".