Thank you very much for your very helpful suggestion! This worked immediately! I would have never guessed.
In case other users encounter the same problem, I quote below the relevant excerpts from the 2-factor identification instructions you sent me:
What if I use two-factor authentication on a device running older software?
If you use two-factor authentication with devices running older OS versions—like an Apple TV (2nd or 3rd generation)—you might be asked to add your six-digit verification code to the end of your password when signing in. Get your verification code
from a trusted device running iOS 9 and later or OS X El Capitan and later, or have it sent to your trusted phone number. Then type your password followed by the six-digit verification code directly into the password field.
Get sent a verification code and sign in with two-factor authentication – Apple Support (UK)