If this story were told from the point of view of Vera, the girl, it would be a completely different story. It would have to have a whole different point to it.
Because the story is told by a third person narrator, it centrers around us (the readers) finding out what has happened to hunting party. The story is interesting because we do not know that Vera is playing a joke on Mr. Nuttel. If the story were told from Vera's point of view, there would be no suspense, at least not of the kind that it does.
If it were told from Vera's point of view, it would have to focus on something else. Perhaps it would focus on Nuttel's mentality and his reactions to the things that Vera said. Perhaps it would focus on Vera's own reasons for telling the tall tale. At any rate, it could not focus on the question of what had happened to the hunters because the narrator would already know.
Saki came to the short story genre as a satirist, and he has outdone himself in "The Open Window." For, by using the third-person narrator, a narrator outside the story, the spontaneous practical joke of the story is played not only upon the unsuspecting Mr. Nuttel, but also upon the reader. With a narrator who is outside the circle of characters, Saki is able to get the readers, who at first laugh at Nuttel's gullibility, to realize that they, too, have been taken in by the tall tale, having had difficulty detecting the truth of a well-told story.
Now, if the story were told from the point of view of Vera, the telling of this fabricated tale would have to be more subjective and the readers might detect its falsity. Thus, the practical joke upon the reader would be lost. And, this practical joke upon the readers is what gives Saki's story its uniqueness and value.
If the story were told from Vera's point of view, it would have been a much different story. As it is, we see that this man is there because the doctor ordered him a treatment of relaxation, and he's visiting these people to give him some socialization on his vacation so he's not sitting in a room alone. If Vera were telling the story, the only thing we'd know is that he's crazy and she's decided she's going to mess with his head.
The ending of the story is powerful because the readers find out that Vera was lying and we find out her attitude towards the entire incident. If the story were told from Vera's point of view from the beginning, the ending wouldn't be a surprise and the whole story would lose a great deal of "power." It would just be a story of a mean little girl, not one of a sick man encountering a child with a vivid imiagination and exceptional skills in using that imagination.