Besides what everyone already said about the events generally being spread out evenly, lore-wise Ul'Dah is the only Eorzean city-state that is actually easy to enter (if you're not a beastman) and do business in that also has a semi-normal government. It was also the home base of The Path of Twelve because that's where Minfilia and her dad went to as Ala Mhigan refugees and where she later grew up, so it makes sense that the story congregates around there during 1.0 and later in ARR when the organization moves to Vesper Bay and rebrands as the Scions of the Seventh Dawn.
Gridania has Hearers who act as bouncers to the Twelveswood, deciding who can enter based on the whims of the Elementals. Before the calamity, the Elementals were doing that themselves with the Hedge and anyone who broke in was killed with treants and wolves. Not exactly the nicest welcome and only select individuals are let in.
Limsa Lominsa has only been a "legitimate nation" for the past 14 years and before then was basically an anarcho-state ruled by a pirate king based on predating on the others around it. Things are better now but the city is still isolated across the ocean from the 3 great continents so it's also often isolated from their problems and we generally only return when we need a boat to go to somewhere else.
Gunbreaker isn't Garlean, it's Bozjan since GNB gunblades don't shoot bullets and we learn it from a Hrothgar.
I think the system we have in place is fine anyway. Expansion jobs come with their respective stones, don't have classes tied to them, and start at higher levels so they feel like proper "prestige" jobs. Beginning the game as a long-forgotten discipline of a past Archon (pictomancer) or a secret Garlean discipline that contracts you with a being of the void to harness their power is a bit over-the-top. The job quests at the correct level for the job all give you the job stones in specific circumstances and it all makes sense in that context instead of randomly showing up at a guild hall anyway.
Leveling MCH/DRK/AST from 30 as soon as the expansion dropped was a giant pain and that was when the cap was only 60 and new content started at 50. I guarantee the vast majority of players do not want to start jobs at level 30 and have to grind all the way up to relevant content.