I had the same issue with my wife’s new phone. (A 13 pro max) tried every solution offered here with no luck.
Because I have multiple Apple TV’s and still
have one physical remote, I was finally able to fix it by performing a factory Reset on the Apple TV. This required then pairing my physical remote with it, and reinstalling apps, but no changes were required on her phone. At that point it just worked.
One quirk is that I performed the initial setup using my phone. That meant that my iCloud account had a generically named “Apple TV” registered along with the “Living Room” device. I just removed it from the iCloud/Apple ID settings page on my phone.