I want to set up a 2nd gen Apple TV, primarily for home sharing, since it was pretty easy to do this with the 1st gen.
Are you sure about that 1st gen? Identify your Apple TV model - Apple Support
The Apple TV (1st generation) [A1218] (2007) hasn’t been useful since May 2018, as it can’t access the Store anymore, and only old versions of the iTunes application (and thus old Macs) can be used to sync files.
The Apple TV 4K (1st to 3rd generation) [A1842 or later] (2017 or later) is much more modern and has different functionalities.
Apple TV (2nd generation) [A1378] (2010) is limited to 720p output, which may or may not be good enough for you.
No dice, even using the trick of generating a 2FA code and putting that after the password didn't get me in. Depending on which menu I login from I get different errors but it's having trouble no matter where I login from.
Signing in for the iTunes Store should still work, afaik. Signing in for iCloud services is no longer supported for this model, so don’t bother. That includes My Photo Stream.
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.