I just updated my phone to 17.0.3 and now FaceTime on my Apple TV works. My profile immediately shows up in the app.
edit: I might have been a little too optimistic. When I click my name nothing happens. There’s supposed to be a notification on my phone but no…
edit #2: It works perfectly on my iPad though, which is on 17.0.2. Weird.
edit #3: If I turn my iPad off I disappear from the list of users on the Apple TV. So… the iPad being updated to 17.0.2 solved the issue. The iPhone being updated to 17.0.3 still has the issue.
I deliberately missed the 17.0.2 on my phone because when it installed on my iPad it got stuck in a boot loop and I had to restore it using iTunes on my computer.
While I did that I also chose to set the iPad up as new instead of using an iCloud backup, to get a fresh start.
To avoid the minimal chance of the boot loop thing happening on my phone I deliberately passed on 17.0.2 and waited for 17.0.3.
Now I’m thinking, maybe restoring the phone completely and setting it up as new again would solve whatever underlying issue is causing this in the first place. But setting up the phone as new again is cumbersome to say the least. I don’t know.
Final edit: I performed a restore on my iPhone using iTunes and then set it up as new, not restoring from iCloud backup. Now it works with FaceTime on AppleTV.
While this may not be an option worth taking just for getting this to work, it might be good to know that the option is still there if Apple takes forever to solve this issue with a software fix.