That's one thing I liked about FFXI. Story missions were a deviation for endgame. Rather than fight a challenging monster for a piece of gear, you fought challenging battles and were rewarded with progressing the story. I hope there are more of these elements present in XIV.
You're right and I absolutely agree the Story Missions were my favorite content in XI by a long shot, and not something you find in a number of other MMOs. I never really thought about them in the context of "end game" because many missions didn't require a character to reach max level to achieve. Also, aside from helping others through the story, you usually didn't see people going through the fights again and again.
The stories were more like precious "events" in your character's career. Something to enjoy thoroughly in your time through, but at the end the time spent on them was far less than the time you'd spend in the content you unlocked after completing the missions. And then you were back to good old endgame grinding.
Not to understate the importance of the Story Missions. Thinking about them after you brought them up makes me realize that it's that content that makes me more excited for XIV than anything else. My time getting through the CoP missions is probably my favorite and most memorable time playing any game of any type, ever. If XIV can give me an experience to that standard, then it will be well worth the wait.