IMO they really just need to add a "postpone" option when you try to escape from a cutscene. Selecting this would kick you out of the cutscene back to your previous quest state rather than send you on to the next as "skip" does.

Something something it's 1.0's fault.

I had this problem as well, until I was told that if you accept the duty you queued for, after you're done the game places you before the cutscene so you've missed nothing. Accepting the duty doesn't skip the rest of the cutscene.
This will work as long as you're using the same job. Since you can't accept a duty as a job different than the one you registered as, and you can't change jobs during a cutscene, if you had changed jobs you either have to eat the penalty or miss the cutscene.