Well, I now can say with some certainty what the status of renting non-Prime content on the Apple TV is, and that is that you can not actually play the video.
This seems odd to me, but the Amazon Prime app allows you to search for non-Prime content and, if you select it, it will tell you that you can play it if you have rented or bought it. You will have to do that on another machine as it will not allow you to do so in the app. If you rent it (and that is all I checked) on a computer and go back to the app, it will say you can play it but, if you try, it will never actually start playing.
I rented a non-Prime video on Amazon and tried to play it 4 times. Each time the app said it was playing it, but it just spun the waiting icon over and over again and never actually did play the video. I could go to my Roku and play the video and, if I stopped it and went to the Apple TV, the app told me it would resume it at the location I stopped at on the Roku, but all it did was spin and spin. It never actually played the video. It seems like a stupid policy, but there it is.
I have not tried this with Amazon content that I bought, only with content I rented, but that would not play and I will have to return the Apple TV to Best Buy.
UPDATE:
Out of desperation, and right before I repacked the Apple TV 4K to return to the store, I rebooted the device (functionality under Settings) and something that the reboot did seemed to have reset the status of the Amazon Prime app. Once the reboot had completed the device did play the rented non-Prime video. The purchase of a second non-Prime video proved that whatever the problem was with my Apple TV 4K, it had been fixed and the device also played the second video.
So my experience with the device refusing to play the non-Prime video content was caused by whatever problem got fixed by the reboot and the device is now working properly.