I had this issue too. And just found a workaround.
It seems that older generation Apple TV’s time-out too quickly before the verification code can be submitted. The solution is to append the verification code to the end of your password right in the same password field. The. AppleTV never asks for the verification code separately. Do this:
- After you enter your Username then type your Password but don’t hit Submit.
- Manually request a verification code on another device: On your iPhone / iPad go to Settings > Tap your name > Password & Security > Get Verification Code
- On your Apple TV, type this code directly into the same password field after your password, no space in between, then submit.
Depending on what screen in AppleTV you started from which it requested your password, you may have to do this twice.
Hope this helps.
Best,
JZKeys