I'm not convinced the Warring Triad are Eikons at all.
Again, this will be purely speculative since FFXIV hasn't said anything about it yet, but the lore of the Warring Triad in FFVI would imply that they Warring Triad are not Eikons themselves; however they may have created the Eikons.
I was pretty young when I played FF VI, but if I remember correctly, during their war between the Triad, the mortals that got caught in the cross fire were turned into Espers (summons, basically) and forced to fight for their masters against one another. This was a "war," after all. The Espers were slaves of the Warring Triad and forced to fight in it. It was not until the Warring Triad had a change of heart and sealed away their own powers that the Espers were set free and given their free will back. If FFXIV is following the same lore as FF VI, than it would follow that The Warring Triad are not Eikons. They would be "True Gods." They do not require followers or worshipers or faith to sustain their existences. They just exist, plain and simple.
Also, if that is the case than the Eikons we know may have originally been tools that the Warring Triad created to war with one another. We do not know the origin of every primal we've faced, however we have learned from Odin, Bahamut, and Shiva, that most (if not all) Primals are twisted versions of real beings that once exist (or may still exist, in the case of Odin). This would fit with the Warring Triad lore. The original process of taking a mortal (Immortal, in the case of Dragons... though they can be killed, so, "half-immortal") and creating a corrupted super-powered version of it was invented by the Warring Triad.
In regards to Odin, it's true that the lore panel states that "he succumb to its power, becoming the Primal we know and love/hate." However, that process is not different from that which iceheart or Louisoux used to summon Shiva or Phoenix. The sword was just using a human host. It's a little different than the other two Eikons we have seen, but not unique or somehow meaning that faith/worship is not required. Odin needed power. He put his faith into the sword when he claimed it as his own, and that was all of the ceremony it needed. Further, we don't know for certain what actually happened to Odin. It's entirely possible that the sword (similar to the way the Warring Triad originally did it) converted his mortal essence into an Esper, and that the phantom we see as Odin is all that is left of him... but it's also just as possible that his real body survived. If Odin had the echo, for example, he may have been corrupted by the blade, but able to regain his true self once he was separated from it. Sadly, we don't know all the details.