The trouble is that defeats the purpose... at that point they may as well simply remove the tombstones from the story dungeons. It is meant to be a mutually beneficial relationship: help players complete the story, get tombstones. However, instead of accepting the tradeoff that they need to wait a few minutes here and there for a cutscene, a huge percentage of players whine and moan about it.
If you want a say over what other players in your party do, form a group outside of the duty finder, and agree to what you are going to do beforehand. In a completely random pairing, it is up to each individual what they want to do, not a democratic vote. If a player is coming to the dungeon to complete the story, and not to get tombstones, it is unreasonable to tell them they need to skip the cutscenes. The duty finder paired you because your task is the same, even if your overall goal is slightly different.
Speed runs are certainly more complicated to deal with, because if the party can't come to an agreement then it's pretty much going to fall apart. I think that one is going to require some changes to dungeon rewards that make it actually beneficial to fight trash mobs. I don't like the idea of skipping through content in dungeons, but with the way that it's currently set up that is still my preferred method, because the "proper" way is sort of a waste of time and gil.
