Not all Beastmen have to summon a Primal but I think the odds are likely it will happen. The Paragons need Eorzea to be in more dysfunction now that the Warriors of Light are returning. Hydaelyn's champions are back on the scene and they haven't done enough in the past 5 years to ensure the City-State downfalls, which probably coincides with the downfall of the Crystals. The Beastmen have needed to be taught how to summon a Primal by the Paragons, but I wonder how the Moogles learned. Moogles are smart enough and isolated enough that they would have fled the Paragons and know them evil.

I still theorize that Fenrir will show up as Menphina's Hound and be an Elder Primal. The Lambs of Dalamud are still active and are obsessed with the Dalamud image (see the ball they created in Tam-Tara). Clearly Bahamut's emergence hasn't stopped them from their goal and obsession with the Twin Moon. And I don't think that all Primals or Elder Primals need to be associated with an element.