Framton would probably not be so gullible if he were not so preoccupied with himself. He can't think of anything or talk about anything but his own feelings and ailments. Vera would perceive that he was that kind of man and that he wasn't really interested in her or even paying much attention to anything she had to say. There is a significant sentence right after they first meet.
"Do you know many of the people round here?" asked the niece, when she judged that they had had sufficient silent communication.
She realizes she is going to have to carry the whole conversation with this diffident stranger and that he is just enduring her presence while waiting for her aunt. No doubt she resents this a little, just as she resents being the one who has to play substitute hostess. It is only because he isn't really listening or trying to carry on a conversation that the idea occurs to her of making up a wild story. She is forced to be saying something in order to break the deadly silence. She may not realize that he story will have such a strong effect on Framton's jittery nerves.
If told from Vera's point of view, the suspense and trickery may have been harder to work in. For instance, the exposition may have had to have gone something like this:
I walked down stairs, showing my advanced degree of self-possession and said, "My aunt will be down presently, Mr. Nuttel." When, glancing at the clock, I realized Mr. Nuttel knew nothing of my aunt or of the history of the area, I knew he would like to hear my aunt's tragic story."
The suspense and revelation of character traits could still be worked in,but the revealing comments would have to come from subtle first-person observations.
If told from the point of view of the girl, "The Open Window" would lose its element of surprise, for without knowing what the girl thinks, the reader only realizes that Vera is telling a story when the Mrs. Sappleton brightens at the sight of the men tramping through the snow. And, the verbal irony in the last sentence would certainly be lost.