Other than Umaro, they ALL have an impact on the story of FFVI.
Relm is Shadow's daughter.
Strago is one of the last of a magical race. (the whole story of FFVI revolves around magic)
Cyan is a major part of the storyline. You even go through his dreams as one of the story points.
Gau is major part of the storyline early on in the game. You can't say any character who isn't an optional character isn't important to the story.
Mog and the moogles save Terra towards the beginning. There is no story without them.
And Gogo is Daryl, Stetzer's long lost love. Also an important part as you go to her tomb during the story when you're getting the 2nd airship.
Relm being Shadow's daughter impacts the story how? That particular subthread is completely optional, not to mention the fact that Shadow is a non-entity after the Continent.
Strago isn't the last of a magical race at all. The Magic City (Thamasa?) is full of magicians like him. These people aren't Esper-Human hybrids either, they're humans who can use magic. He doesn't actually do anything besides tell you where the statues are, and, I think the Gate? But the Empire already knew the locations of both of those, so whoops!
Cyan stops having any impact on the story by the time Sabin finishes his little scenario. The Dream Demon? That (optional quest) only tries to flesh out Cyan's incredibly complex backstory of "Being pissed that his wife and son are dead," which is what you find out within 10 minutes of meeting him (him being Cyan).
Gau gives you the helmet, and like Cyan, stops having any impact on the story by the time Sabin finishes his little scenario.
If you're going to try and say that Mog is important because of that, that means all his friends are just as important to the story, which is kind of silly, especially since that scenario could have played out with just Locke, really.
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Any RPGs with various playable characters?