The eye, but not the brain, takes multiple shots with different exposures. That's the purpose of the eye's iris that acts like a diaphragm to limit the light. As long as the iris stay the same, the brain gets the same exposure for all parts of what the eye sees. The parts that are not focused on do not change in brightness as long as the diaphragm stays the same. Even if you were to focus on a shadow area and allow the iris to change, it mostly does not show the details within the shadow.
When you use HDR to open the shadows, you're seeing something in the photo that the eye isn't ordinarily seeing. You are compressing the range so that the darkest darks and lightest lights are closer together than the eye and brain sees. HDR overdoes the lighting range by compressing it beyond what the eye itself can do.That's why HDR shots look unnatural.
Alan, I can only speak here of what my own eyes see. Maybe your eyes are different. The most obvious example of what I'm talking about here, is the view out of a typical living room on a bright day, as you sit at the far end of the room furthest away from the window.
Provided you are not attempting to look directly at the sun, you should be able to clearly discern detail in the white clouds and appreciate the rich blue of the sky. Redirect your gaze to any relatively dark area of the room, perhaps the lower shelf of a bookcase, and your pupils should immediately dilate, allowing you to clearly see the contents of the bookcase.
Take a photo of your room with a lens wide enough to include both the bookcase and the view out of the window, and you should find that whatever exposure you use, you will not be able to capture the detail that the eye has seen in both the bright white clouds, and the contents of the bookcase in the shadows. If you don't believe me, try it.
If you have a camera with a relatively good DR, such as a D7000 or D800, you might be able to expose for the sky and raise the shadows in post-processing so that certain large items in the bookcase are vaguely discernible, such as the main titles on books. But such detail, if visible, will be surrounded by all sorts noise which the eye just doesn't see in reality.
In order to capture closely and realistically what the eye has seen in such a scene, it is necessary to take at least two shots with the camera, for purposes of creating an HDR image. The HDR image, if processed skillfully and properly, will more closely represent reality than the single shot.
The fact that sometimes HDR images are not processed properly or expertly, and result in an 'unreal' impression, perhaps due to the application of a formulaic process of so-called tone-mapping, is another issue. All images have to be processed in one way or another. If your HDR image doesn't look right, then I recommend identifying why it doesn't look right, then fix it.