So I've been stewing over this a bit and I think I can tie each elemental primal to a deity, specifcially the six that are not tied to a city-state. Although, it requires a little gender-bending, although we could put that down to lore convolution over time. "Mired by a plague of doubtful and cheap lore."

Fire:
Nald'thal is guardian of Ul'dah.
Azeyma the Warden represents the sun. It is more of a searing and powerful heat, which is very much Ifrit's essence. Ifrit even emerges from a miniture sun when summoned. Ifrit is also rather protective of his Amalja tribe and devotees, very much like a warden.
It is not the more subtle fires associated with the kindling fire of life or the torturing fire of death.

Wind:
Llymlaen is guardian of Limsa.
Garuda is characterised by being wild and free. She will not be bound. Everything about her involves the freedom to move and the Ixal worship her for what she offers them in their pursuit of flight. This resembles Oschon, the Wanderer, who roams the land.

Earth:
Nophica is the guardian of Gridania.
Althyk, the keeper, is master of time, which is a mark of endurance. While we have little to go on here, we know that Titan and the Kobolds are based in the depths of O'ghomoro where they are steeped in tradition. Their patriachs may become outcast if they leave the mountain. This focus on timelessness ties well to Althyk.

Lightning:
Rhalgr is guardian of Ala Mhigo.
By comparison, Byregot the builder is reflected in the nature of the Sylphs, who seek the building of peaceful relationships. Their affiliation to lightning can be seen in the emblem at their altars, so I believe wise old Ramuh would be their primal. I'm joining limited dots here. The Sylphs, in their wisdom, are fearful of calling on the power of their primal.

Water:
Thaliak is guardian of Sharlayan.
Nymia the spinner of fate has connections to shrouds of mystery. Everything we have learnt of Leviathan is very mysterious and so are these fabled Sahagin. People fear them and their unpredictability.

Ice:
Halone is guardian of Ishgard.
In her many incarnations, Shiva has always been some form of seductive woman, which aligns with Menphina, the lover. I'm unsure what beast tribe would summon her.

Yes, I've left out Odin and Bahamut, mostly because their associations usually go beyond base magic and into something greater.