When you enlarge the photo and display it at the field of view it was captured at, as you can do in VR, the whole problem disappears and the photo just looks like the real world. Use two cameras and it'll even be in 3D.
If you intentionally use a nonlinear perspective like the Turner painting in the article, you can get an image that looks great flat, and still produces a wide angle of view and "natural" dense of depth with low perceived distortion, and will look fine from across the room or up close. It probably does a better job of recreating what it is like to be there than a mechanical linear perspective photograph. The only problem is you can't get images like that out of a camera.
That's true, but not the problem. Think about this logically. The goal is to show the exact same color at the same angle. If you achieve that (possible with VR, also possible with a normal photo but for a tiny angle only), it doesn't matter how the eye or processing in the brain warps the image afterwards.
You can throw away the straight lines and shoot with a fisheye lens (no straight lines, wide angle, fairly natural depth), you can crop in (depth fine, but now I'm missing stuff around the edges), and you can shoot with a wide rectilinear lens (wide angle, straight lines, "exaggerated" perspective).
As far as I know, the answer is simply that your eyes aren't rectilinear but your brain fudges it to make straight lines seem straight when the image on our retina actually curves.
The effect is most prominent when standing in a small room. It's impossible to take a single picture of it that conveys what you want: a wide rectilinear lens will exaggerate how big the room is (and look weird around the edges), a "normal" rectilinear lens will be so cropped in that you can't see most of the room, and a fisheye lens will make all straight lines curve, especially around the edges. None of them really represent the visual experience of being in the room. Just look at how weird real-estate listings can get when they try to show you a picture of a small bathroom.