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You made some good points. Not all I agree with, but they are no less good points.
Anyways, my answer is a lore-based one. If you don't care about the lore, then really what I'm going to say is a moot point, and this would be the time to stop reading here.
Now then, you're primarily an Adventurer, right? I mean, sure, you're a now-famous, primal-killing, case-solving, craft-making, high-ranking Grand Company Officer/Scion/diplomat/Warrior of Light, but still essentially an Adventurer at the core, right?
Dungeons are adventures. They're the stories you tell/are told in. The stuff of legends. Rumors on the streets of Ul'dah and drunken tavern tales in Limsa Lominsa. They are one of the reasons your character is so famous and prestigious in Eorzea. They are what made you. Dungeons are the road you traveled to greatness.
If the experience of the game overall is the forest, dungeons are some of the trees. Don't miss the trees for the forest.
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Dungeons, in basically every MMO I've ever seen, are a prequel to raids. They're easier, they're smaller, and they give weaker gear. This is what they do, they help new 50's gear up to actually attempt higher stuff, they help players gear their alts, and they help bridge the gear gap at each new large update.
The problem isn't that dungeons are pointless, the problem is that Hunts seem to outweigh everything else on the reward scale. This wasn't an issue until Hunts appeared, as you couldn't get myth or gear through any other means. Dungeons dropped gear and tomes, which you used to buy better gear before moving into raiding. FATE's sure as heck aren't useful toward that end, neither is crafting, nor treasure hunts... etc.
Dungeons are still right where they belong, once SE fixes the progression ladder.
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Dungeons are where crafters get their mats. Duh.