But that's true for FPS, too. The AI is braindead. You play not for the challenge but the setpieces, world building, crazy action scenes, and the story.
Eh, that's not really true for every case. The Halo games have AI that at least feels like you're playing against characters that are on equal footing with you. That and the stages provide a variety of options on how to approach situations.
The "1-on-1 with a static arena" nature kind of already limits what you can do as far as scenes and set pieces go and unless you have decent AI (which, let's face it, fighting games don't really) there's not much to see outside of the core combat.
please tell me what is qualitatively different about a fighting game's gameplay compared against, say, a chatacter action game which prevents it from being able to have a narrative.
If we're talking about 1-on-1 fighting games, you can do cutscenes but environmental storytelling is quite a bit harder when the way you move through arenas is limited. Of course you could do a Tekken style Evil Within style mode but then you're making a completely different game in addition to the fighting game. Which is completely fine as an approach if you're willing to commit to it but Tekken 6 shows how that can go wrong.