Two things not yet mentioned I feel it might be worth contributing.
1: As we have learned from Leviathan and the Whorleater, and to a lesser extent Garuda, corrupted crystals have an effect on Primals. Namely they can disrupt or even disable their control over aether. And what is Bahamut's physical body down in the coil almost completely covered in? Could that be by design?
2: If Elder primals are, as their names would imply, the oldest and most powerful of the primals, then it might stand to reason that they had to be called into being by older, more powerful beings themselves. I also find it unique that there are specifically TWO such Elder primals. Coincidentally the game also has TWO very old, very powerful beings in its lore: Hydalaen and Zodiark.
Suppose for the sake of argument that Zodiark itself "summoned" Bahamut. That would explain:
The Ascians fascination with him.
His lack of need for crystals or worshippers.
His unrelenting hostility towards Eorzea.
His penchant for causing GLOBAL mass destruction, as Hydalaen HERSELF is his primary enemy.
His singling out of Louisoix - Hydalaens former primary ally as the Archon.
Furthermore, this would lend credence to the theory that the crystal of his heart is, or is a piece of, Zodiark himself. SE has a notably similar event in their company lore - Yu Yevon, a magical entity who summoned the colossal and destructive being Sin to serve as his body and protector.
So what about Odin?
Well it would stand to reason that is Bahamut were summoned by Zodiark, that Odin would have been summoned by Hydalaen. Lets look at some piece of that puzzle:
What is Hydalaen's modus operandi? She:
Chooses one person to be her crystal bearer.
Empowers that person to stand against the darkness (Zodiark and his powers).
The chosen person becomes extraordinarily powerful (to the point that our player character "makes sport of killing gods" as Merlwyb phrased it)
Odin makes an implication of his present or former humanity.
He takes the mortal form of whoever killed him last.
That last line there seems to be the strongest indication of my theory. What if rather than a mere game mechanic (bragging rights of the kill), the change of his form is actually LORE based. A sort of warning. "I become like you" can also imply "You become like me." Or "In some unnamed way, we are, were, or will be alike."
Assuming this is true, then it might fit that Odin was once some incredibly ancient champion of Hydalaen herself. Perhaps initially a Crystal Bearer. In order not to lose her champion after his mortal death, Hydalaen herself summoned him back as what we would call an Elder Primal. One that would remain sealed in a place of Hydalaen's natural splendor (the lush forests of Gridania), and be released to defend the world in the event of either great catastrophe, or perhaps in reaction to Bahamut specifically. Again, there is SE precedence for such a being: WEAPON. So why would it be that Odin merely wanders around, waiting to be fought? It might simply be that, as we are so often told, Hydalaen is weak and the last of her strength is waning. Odin was released by the Calamity, but she is now too weak and unable to control him in his Primal state. So she needed a NEW champion. One who could act of their own free will, not needing her power to control.
I'm also willing to bet, that like how Bahamut has a physical form that is sealed away (in the Coil), and a "manifestation" form (what was sealed within Dalamud), we have only too seen Odin's manifestation. Somewhere, probably within the Shroud, is his physical body, entombed by Hydaelaen.