Apple TV box cannot sign on

If your Apple TV box is really 1st generation (white, 7.7"), then it would indeed no longer support connecting to the iTunes Store. It has been like that since May 2018. It is functionally obsolete.

If your Apple TV box is black and 3.9", then it is at least 2nd generation.

Identify your Apple TV model - Apple Support

I’m sorry to hear that; it worked for so many others. For further assistance, I would suggest to contact an expert from the Apple Support staff. Contact Apple for support and service (phone numbers) or start your support request online (chat/​email/​callback).

If you have an Apple TV (2nd or 3rd generation) and two-factor authentication for your Apple ID, 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 passed the time limit.