Well, eikons are primals, but still, Odin is an exception. Which "mortal desire" does he represents ? It's not even the one of the guy who takes its sword, because it's the guy who becomes Odin, not the other way around, like for Ysayle.
Right now, we don't know enough about the warring triad... We can see their statues though, and one of them seems to have a third ete like garleans but... What are they ? Don't know.
Primals are power incarnated through crystals and will, but maybe there are "true gods" like Odin and the warring triad, That behave in a similar way... Still, that's not convincing, but that's only what we've got right now. Maybe the warring triad are Hydaelyn, Zodiark and Midgardsormr locked in a human-similar body, maybe they are mortals That became so powerful they became gods (like Odin is assumed to have), but really...
I really like to speculate on lore theories, but I don't feel like doing it with so little x)