Solved!!! ... at least for me.
1) From Apple TV: Settings -> Users and Accounts -> Add New User
2) Move your iPhone close to your Apple TV
3) From Apple TV: Select the option to "Sign in with iPhone or iPad" (if nothing happens, lock and then unlock your phone)
4) On your iPhone: Select to Connect to the Apple TV
5) Wait while keeping your iPhone close to the Apple TV
6) On Apple TV: Make sure that your account added completely (sometimes there's an iCloud or access issue) and then back all the way out of the Settings.
7) From Apple TV: Settings -> System -> Restart (ie NOT RESET) ... why this is needed, I don't know, but it seems to help
8) When the Apple TV starts back up, go to the FaceTime app.
9) On the FaceTime app's Continuity Camera screen, your account SHOULD be listed as a User option. Select it.
10) It might take a few seconds, but when prompted on your iPhone, accept the Continuity Camera option.