Actualllyyy, now that I think of it, I believe the Binding Coil of Bahamut (the first set of 8-man raids, used to be endgame) contains what i consider to be significant story.
I'm unsure who does these old raids anymore, and how they would react to someone watching cutscenes.
I assume you would attempt these raids with your friends, so even if you get uppity people from the Duty Finder, a few of you will be watching cutscenes.
And it's not like a standard DF group could easily 4-5 man these fights anyway, so they would have to wait for you >

.