It’s because the retina — where light is processed into signals for the brain — it’s kind of like a movie screen — is not immediately adjacent to the iris.
If the retina were adjacent to the iris, then yes, a smaller iris aperture would mean a smaller field of vision.
But light hits the retina (the curved movie screen) pretty much all over whether the iris aperture is tiny or large, because the retina is a centimeter or so away from the aperture.
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