To answer the question posed in the thread's title; well yes, but actually no.
An MMO needs a story, and it needs a really big and intricate one at that. It needs to be bigger than the game world, and the game world of a successful MMO needs to be very large in most cases. However, it needs to be an MMO story, and not an RPG story. What I mean is that the story in an RPG is typically driven by the characters in the game and how they overcome some set of seemingly insurmountable obstacles. However, for an MMO, such a story is at odds with the MM part of MMO.
For an MMO it is much better to write a story that is not character driven, but rather world driven. This is a very different type of storytelling, where the events of the world are what take center stage. The best example I can think of for this type of story telling is ironically not an MMO, but rather the Souls series. In a Souls game the world is the story. You don't experience the narrative through how your character grows as a person, but rather through learning about the world and its events. You experience the story through learning about the world and how the characters interact with it. The stories that involve characters aren't about how the character grows to overcome obstacles, but rather about showing how people interact with the challenges of the world. The progression element in a world centered story comes from the knowledge the player gains as they experience the world, not how characters grow.
TL;DR MMOs have the potential for amazing stories, but XIV's doesn't really utilize this potential.
Below is a writeup of how ARR could be changed to be a more world driven story, to give you an idea of what such a story might look like in XIV's case.
You could have a more world focused story in ARR just by removing the player from the chosen one/main protag role. Instead you would remain a simple adventurer. Gifted with the echo perhaps but not chosen by Hydaelyn. You'd still come across the scions, but instead of them becoming the "main cast" they'd have their own narrative unrelated to the player. This would mostly be focused around how they work towards their goals of eliminating the primal and Ascian threat, with the player helping out here and there. However the player wouldn't be tied the Scions and would spend much more time dealing with the Garlean threat than they do in the current ARR narrative.
This shift in narrative would allow for the player to learn more about how Garlemald affects Eorzea early on in ARR. During the course of these adventures the player would of course also have ample opportunity to take time to help the common folk of Eorzea, and in the course of doing so could choose to learn more about what ails them and how they live their lives. This would create a much more down to earth narrative.
These changes would of course kill some of the story momentum without some minor character adjustments. So naturally the competence level of the Scions and the City State Leaders would have to be bumped up a bit to provide that, and Garlemald would have to be more of an active threat.