I have to admit I have one big issue with your question, and that is that I do not think that Framton Nuttel is actually the main character. Although we certainly start with seeing his perspective in the story, it very quickly becomes clear that the main character is the storyteller par excellence, Vera, who is able to deceive and trick Mr. Nuttel with consummate ease and then does the same to her family.
And this really is the reason why Saki chooses to bestow upon Framton Nuttel this "nervous condition." This explains his susceptibility to the story that Vera weaves and explains his quite sudden and shocking reaction when he sees the "ghosts" return from their hunt. As one of the returning hunters comments, when he saw them, Mr. Nuttel "bolted" out of the house straight away, allowing Mrs. Sappleton the opportunity to call him an "extraordinary man."
Thus Saki gives Mr. Nuttel his "nervous condition" to show how open he is to being deceived and also perhaps to question Vera's intentions. On the one hand this is an incredibly amusing story, yet seen from a different light, it is an exercise in cruelty.