@Eric I wanted to call this out in a recent thread on this topic but it's closed. I read the thread as I have recently seen the disk full message on my Apple TVs. At the end of the thread you linked to TestFlight to sign up and download the beta app as you believed this would no longer cause this error. I tested the beta app yesterday and I was watching live tv when the stream froze and the app was not responsive. I went to settings, general, manage storage and I saw that tthe channels beta app showed 7,35GB of storage in use. The Apple TV apparently ran out of space, did not throw the disk full error but the unit pretty much locked up. I rebooted and moved on to use the unit again. I thought I remembered reading a max usage of something like 1.8GB or something like that but this was definately much more data used.
Please submit diagnostics from the app after this happens.