You won't see the aperture change until a picture is taken. Because the camera uses through-the-lens metering, the camera keeps the aperture at a wide aperture setting (usually the widest possible). This allows the most light into the lens for more accurate metering.

Additionally, the shallower depth of field with a wide aperture helps the autofocus achieve sharper focus, because areas in in front of or behind the plane of focus fall out of focus much quicker with a wide aperture.

When you press the shutter button, the aperture iris will close down to the determined or chosen value, and then the shutter will actuate and capture the image. After the shutter closes, the aperture resets back to the widest setting.

I noticed another strange behavior after closely investigating this: as I screw the lens on, the further I screw it on, the wider the minimum aperture becomes. I screwed it on while the aperture ring on the lens was at f/22, and the more I screwed it on the wider it became even though I wasn't touching the aperture ring and I verified that I was not moving it.

This is expected behavior. The aperture lever on the lens is spring-loaded, and rests against the aperture control linkage on the camera. The default position of the aperture iris (when the lens is not attached to the camera) is to be as small as the aperture setting on the aperture control ring, or for electronic lenses without an aperture control ring, the smallest possible aperture. As the lens is rotated into its secured, affixed position in the bayonet mount, the aperture is being opened up to the (usually widest) aperture that the camera uses to meter and autofocus.

The camera says its changing but the blades don't physically move and I've even tested while taking a picture.

No, it doesn't say it's changing the aperture size. The camera is noting your commanded (or the program-chosen) aperture setting, and the exposure meter is adjusting its reported value according to the aperture it will use during capturing the image, not during composition.

Regardless of what I set the aperture to be on the camera, the blades move to the same location, about f/5.6, when I click the shutter. I tested this by taking a 2 second exposure so I could clearly see it.

I believe the aperture should open up to the lens's widest setting (f/2 in your case); but it is possible the D500, or conditions that it is being used in, only open up the exposure to f/5.6. But I wonder, how sure are you it's f/5.6 and not f/2?

If the camera is trying to open the lens to f/2 during composition and metering, but the lens only goes to f/5.6, then there might be something wrong with the lens. However, everything else in your question is expected behavior; it is merely a misunderstanding of how the physical aperture linkage system works on the Nikon F-mount bodies and lenses.