The main problem I can see with such a system is inevitably players would still find the one that gave the 'best results' or give the best benefit for the least effort and then everyone would simply go for that. It's why such systems of 'choice' usually do not work very well in MMORPGs because 'choice' is misleading if not illusory.
Also it is increasingly clear in the game's very story that the Twelve... do not really exist (at least not in any way that directly interacts with the mortal world). It's Hydaelyn Herself (as in the Mothercrystal) that is the very real deity that is directly influencing the player's character's life and story (after all, She is the one that gave the player the Echo, and by extension, the Blessing of Light which is what empowers them as the Warrior of Light - gameplay wise manifesting in the 'Power of the Echo' to 'transcend your mortal limits' by offering stat boosts in battle).
So I think the Twelve will continue to fade into obscurity as Hydaelyn takes over as the main deity in Eorzea (and thus is also used in gameplay). Your idea about using the Twelve has merit, but the whole use of 'junctioning' members of the Twelve... does not really fit FFXIV's setting (the Garleans after all regard the Twelve as 'false gods' after all).
Having said that, it appears at least one point in 1.0 that the game's story was leading up to a possibility of maybe being able to equip a Primal in such a way like the Totemas from FF Tactics Advance - Ifrit basically told the player's character outright that as "one of the godless Blessed" (as in, an Echo user), they can summon their own Primal and forced them to swear that they will never do that - the player's Path Companion ponders that possibility only to then dismiss it as a "taboo best left unbroken."
As this line was completely scrapped from ARR it's pretty clear SE have long since abandoned such an idea alas, so I do not see SE using such a system given how SMN was executed with summoning Egis, and my earlier comment about human laziness just taking over and people simply chosing what gives them the best benefits for the least effort.