I disagree that the first six games have little to no story, unless you are using some ass-backwards definition of what constitutes a story to back your argument. I also don't really subscribe to your idea of "moments over story" definition of FF. Because we continue to play for the story and gameplay, not for the moments. I think you're letting the PSX generation use of FMV cutscenes warp your brain on what constitutes as rewards in the FF experience.

Take FFIV for instance. All but the most die-hard fanboy/girl of this title will openly admit that FFIV's plot is pretty ridiculous and so fantastically over the top with some of it's drama that its more funny than serious (Cid's "sacrifice" is probably the most easy scene to see this in action) but I feel the story and narrative are kept fresh due to the writer's excellent crafting of Golbez in the scenario. IV's plot is about the war, not the individual battles (moments) that keep you playing. When Cecil becomes a Paladin, you don't keep playing because you hope to see a cool scene like that again, you keep playing because the writer has done a good job of making you dislike Golbez and hoping to to see him to injustice, with all the interactions you've had with him proceeding this moment. Moments are what us old players keep playing the older games for, so we can re-experience the joyful nostalgia of the first time we tried to devour the games narrative and contemplate the existence of it's rich cast of characters. Moments are simply the best part of the story that stick out to us old fogeys, and make us want to replay them, but I would argue that watching the scene on Youtube is not the same as actually playing through the game. Story, is why we first play the game in the first place. We wouldn't even pick up the title if we didn't know what the premise was, but it is also what binds us to the moment as the narrative builds and we experience the flow and pacing to the title that leads us to this defining moment.

Bringing this back to FFIV as example again, why is the Paladin transformation scene such a powerful moment for the player? Because of way the story has been building up to this moment. We witness Cecil's rebirth, not just just literally but figuratively as his new journey as a Paladin begins where players were first introduced to Cecil in the first place, through his connection to Mysidia. Cecil has come full circle by this point, and the player is happy to finally get a moment of triumph because up until this point, Cecil and the player has been experiencing nothing but failure and defeat through the hands of Golbez. The Paladin "moment" is only powerful because the good writing of the story actually makes us care.

My issue with your ideological premise is that it's completely backwards. Moments don't make the story, the story creates the moments, because what is a moment without context that is given by the story? Not much, so how can these define the greatness of a narrative if they can exist without context, subtext, and emotional investment from the person experiencing it?

The FF series is not exactly the most well written series but it doesn't mean the series overall has bad writing, and when something like FFIV can still be in the hearts of players long past an era with better production values and memory space to create greater context and better utilize subtext, then obviously the story must be doing something right to be so beloved.