Apparently, current flowing from the REMOTE port to the ESP32’s GPIO pin was to blame, because adding a diode in between fixed the problem. The GitHub repository holds the design files and code.

The only noticeable difference between the two is a "mute" button on the TV remote compared to the voice remote (voice control still works for muting the television, by the way).

In iOS 11 and later, it's possible to add an Apple TV Remote interface to the Control Center that will allow you to quickly navigate and control playback on your 4K Apple TV or fourth-generation ...

Microsoft Remote Desktop Connection: Setup For users with computer chops, setup and use of RDC is no big deal—remote desktop control isn’t rocket science.