With many fisheye lenses the distortion will NOT look normal viewed close up (I'm not sure any do) There are multiple fisheye designs which map the view in different ways (the mapping function section of en.wikipedia.org/wiki/Fisheye_lens shows this well), some of my 'fisheye' converters cram a significant part of their FOV into the very edges of the image. I think the 'Equisolid angle' function is the nearest for these. Fortunately most of my true fisheye lenses are closer to equidistant or stereographic designs & give a much better image. However fisheye lenses are a rather special case, using distortion to fit in more than would normally be possible!
Changes to the viewing conditions are not the perspective of the print, they relate to a viewing perspective that is not part of the print. There are certainly conditions where the viewing perspective can correct for distortion of perspective in an image. For rectilinear lenses Ansel's claim that perspective of a print is dependant only on camera-subject distance is 100% correct. Not a fallacy as the OP claimed.