Failing to Connect Remote App to AppleTV (4th Gen), While Screen Sharing Works

I don't have access to my proper AppleTV remote to control my AppleTV (4th Gen), but that's no problem because I can use the Remote app on an iPhone and iPad. The Remote app was working fine until one morning it wouldn't connect from any iPhone or iPad. The status listed is "offline" -- see the attached screenshot. The AppleTV appears to be still connected to the WiFi network, as I am able to share my iPhone's screen. How would screen sharing work but the Remote not work? I killed the Remote app multiple times (swipe up) and rebooted the AppleTV (power cycle).