If you have an Apple TV (2nd or 3rd generation) and two-factor authentication for your Apple Account, then you may be asked to type your password followed by the six-digit verification code directly into the password field. Additionally, there is a time limit, after which the code becomes invalid.
Try this: First enter the characters of your password (do not submit yet), then manually request a new verification code, then append that code to the password (no spaces or other separating characters), and then submit. That should help you get past the time limit.
Please let us know if that still works in 2024.
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Note: You should be able to sign in as described for iTunes Store services. You cannot sign in for iCloud services anymore on these legacy model devices. iCloud has changed, but the legacy software hasn’t changed with it.