Many of us do. Speaking for myself, the storyline is the only game-related component that I stick around for. That said... a few other notes for me.
(1) I don't think the quality of the storyline arises from its continuous nature. FFXI had storytelling that matched or exceeded XIV, in my estimation, and that game split it into parallel chunks. I think the storytelling quality arises from the fact that SE has a good development team for it, albeit their writers use about 50% more words than necessary at times.
(2) The storyline does very little for my day-to-day immersion. It simply isn't linked to the game world very significantly. While I'm doing main scenario quests, sure, I'm invested - but as soon as they're done, they go on the back burner again. This isn't a complaint or critique directed at SE, but it's for this reason that I disagree, to a large extent, with your assertion that FFXIV's players have such a great thing on their hands. Storylines are quite compartmentalized in MMOs, even FFXIV; good ones add to the experience, sure, but it doesn't make up for other lacks.
(3) I also disagree with the assertion that FFXIV is about story. Or rather, I think it's selectively* about story. When new expansions launch, I agree 100% - and I don't think it's any coincidence that these initial launch periods draw so many players back to the game for awhile. Afterward, however, the story takes a back seat. I mean, really, we get what, a few hours of main scenario quests every three months? That's a pretty light touch. FFXIV is hardly a story-driven narrative right now, for example. There's no feeling of 'story' when grinding Roulettes, or crafting, or gardening, or messing around in Eureka.
Finally, and this is certainly a matter of perspective, but I don't hold XIV's story up on much of a pedestal, personally. It's good by MMO standards, but it's average at best when put up against single-player titles. The impact it has on individual players really varies as a result. I am glad that you enjoy the storyline, and find FFXIV immersive! Just realize that its quality is very much in the eyes of the beholder, even from a storyline perspective.