If we count Palace of the Dead as a dungeon(its... special), there are 6 dungeons you can unlock without progressing the story*. The MSQ for the base game(not counting expansions) will take you to(and thus unlock) 11 dungeons. Additionally, it gates access to 15 more optional dungeons behind completing a specific quest(the one that ended the original 2.0 release of the game). The MSQ will also send you through 3 mini-trials and 10 actual trials(big boss fights without an attached dungeon) - all optional trials are also gated by completing the base game(as are both sets of raids). All of this is my way of saying, if youve been enjoying the dungeons(and similar content) you get access to a lot more of them through the MSQ then without it.

As for the MSQ itself, it starts slow. The MSQ tells the story of how your character went from a wandering adventurer no one knew to the savior of the world. How does that happen? You start helping out with small problems, eventually get noticed by some big fish, asked to tackle larger problems, and so on until the fate of the world(literally it turns out) depends on your actions. Which is to say, the story gets better. There are peaks, there are lulls, but basically everyone agrees it gets better. And yes, it has been announced that the next patch(currently shooting for mid-late july) will include some changes to the MSQ. Dont think of this as a magic bullet. Theyre making some trims and adjustments, but we've been told itll amount to something like a 13% reduction.

*Unlocking Palace of the Dead itself requires you to go a little into the story. I believe it becomes an option once the MSQ has unlocked its 3rd dungeon.