I read your whole post, and I won't comment on the rest of it because this is where we disagree and we can't get past that:

First lets go with a checklist the big things that summons over the years have had in common for the thematic side. Good appearances(animation/looks), strong iconic moves, the entity is as strong or stronger than the character, borrow their power for a limited time, and often times have to earn their use. (For those not familiar, appearances can be extremely subjective, though usually the FF summons are a step above most of the other spells/actions/monsters in the game.)
"Strong" iconic moves. I hope you mean "strong" in the visual sense.
They have never really been stronger than the character, barring XI. They're either an MP sponge(sustainability problem) or there's just better options.
Limited time really has nothing to do with the theme, and X didn't have a timer. XI didn't really have a timer at endgame, and Carbuncle especially didn't.
Earning use? Yeah.

XIV hits all those notes for me, though.
"Strong" iconic move -> Enkindle
Not insignificant, damage wise, compared to other Final Fantasies. This is due to them being permanent and aiding our character. No single hit is that high, barring Enkindle, but all the smaller numbers add up.
You earn the use of Ifrit/Garuda/Titan.