Cannot Use Airplay 2 or Home on Apple TV

Just updated to TVos13. When I went into "Settings" on my Apple TV, there were two red alert badges.

One, under "Users and Accounts", led to a menu item to "Update Apple ID Settings". This led to a screen asking me to "Add this Apple TV to your Home" by unlocking my iOS device, turning on Bluetooth and holding the device close to the TV. When I do this, nothing happens. I've tried using an iPad and an iPhone, restarting the iOS devices and the Apple TV, switching WiFi networks, and nothing.

The other alert badge, under "Airplay and Homekit", leads to menu item to "Finish Home Set Up". This leads to a screen asking me to "Start Using Airplay 2". The instructions are exactly the same - unlock iPhone, turn on Bluetooth, approach Apple TV - and, again, absolutely nothing happens.

The Apple TV never appears on the iOS devices Bluetooth screen, so maybe it's a Bluetooth problem? Any workarounds mutely appreciated.