If you knew what deus ex was, you would know why Hope wouldn't qualify- that whole being involved throughout story thing. Deus ex is the appearance of a new character (event, object, etc) that provides a solution for an unsolvable problem.
While the act of the appearance with the calvary might be one, that was not Hope's primary role in the story.
And your other argument makes no sense. If it wasn't Sazh it would've been another pilot who was a deus ex- so what? How does that make him not one? X was one, but if it happened to be Y it would still be one therefore X wasn't one...
I said I was done last post and I am after this one- but since you didn't know what a deus ex is I figure one more post on this isn't gonna kill me.