Does the older apple tv still work??

Apple TV (2nd generation) is from 2010, and has but a fraction of the functionality of the later models.

But you should still be able to use it for what it does.

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.

… I called Apple support. They could not help me […] since it was 12 years old …

Yeah. The very old models receive no official support anymore.

So I bought a new Apple TV.

Awesome. I think you’ll enjoy it very much.

Apple TV user guide for tvOS 15 - Apple Support

for Apple TV (4h generation or later).