I appreciate the help and advice. I got a new tv within a few days of getting the 4K Apple TV, so I don’t know whether the tv, the HDMI, or the Apple TV is the problem (the same HDMI cable was connected to the previous gen Apple TV and I never had any issues, and I didn’t notice a problem with the Apple TV 4K before upgrading to a 4K set, but it had only been a day or two before I switched) and am trying to save myself a lot of troubleshooting time.
At least now I know that replacing the Apple TV won’t solve the problem...