I get the frustration. To see all this cool loot show up, but it's maybe not for your class or you roll low and you get nothing. But if you got a piece of loot for your current class with every chest or set of chests in a dungeon, you would get a full set of gear in 1.5 runs of the same dungeon. That just doesn't seem like something they'd do. It's too easy.
But as some other folks said, if the loot is for your class, hit Need and it's yours.
And yes, you'll have reason to re-play those dungeons as you level up. In later levels, you get access to Dungeon Roulette (and eventually other systems), which gives you an exp bonus for essentially helping lower level players to get through the story. It's very likely that the three people in your party were much higher level, but their levels had been synched down to 15. They got a bonus for helping you.
And they'll probably let you have every piece of loot in every dungeon you run up through lvl 50. Even if you can't equip it, you will unlock systems that give you rewards for loot you can't use.
I would say that a lot of people will wait for you to watch the cutscenes. Maybe not most, but a LOT of folks I've played with are nice like that. The only dungeons I had trouble with in this regard were the last two in the ARR storyline, and it was a bummer.
But you can go back to your Inn room and watch the cutscenes there if you miss anything. It's less than ideal, but it's a nice feature.