The reason the Apple forums discussion of the issue, which I linked to in post number 6, is twenty-four pages long, is because it really isn't as simple as you suggest.
The situation for me occurred as follows: About 10 days ago I finally decided to adopt Apple's two-factor authentication (which in general is extremely well-implemented and seamless.) I think I updated credentials on about 10 computers several iOS devices without any issue.
Several days later my wife uses our Apple TV2 to navigate to a David Attenborough documentary we had purchased several years ago, and when she hits the play button, we get prompted to enter my Apple ID and password, which I do. Then a second pop-up window appears, asking for a two-factor validation code (which simultaneously appears on my other devices, exactly as expected). The problem is, there is no field in which to input the number, and eventually I read on the link you posted that it can be done, but one must immediately append it to the end of the password. So I tried that about 2 dozen times, without success. It just goes into an infinite loop of rejecting the password followed by the above behavior.
The trick is on page 24 of the discussion linked in post #6. Unfortunately, I was only on page 2 at that point, where one subset of users was talking past the other subset. The trick, which I described in post #10, works, but it is not intuitive. First, you have to recognize that the prompt sequence I was getting will never work, and you have to access the interface differently.
Like you, I logged in with my credentials the day I bought the device, and haven't had to look back. This is precisely why it hadn't occurred to me that now I needed to do that (after first signing out).
What finally worked was to go to the iTunes Store section within the Settings app, force a log-out, and then re-log-in, using a new password that is a catenation of the old password and a fresh 2-factor authentication code immediately generated on the spot from within System Preferences>iCloud>Account Details>Security. What pops up on the screen of your iphone not only won't work, but it will sabotage the process. This procedure is not intuitive, it is not described by Apple, and is distinctly separate from what appears on the ATV2 interface by default.
Several folks here use "legacy" ATV2 and ATV3 as audio streamer/zone player devices (a "poor-man's Sonos"). I first thought one of them might have run into the problem and would have a solution. Hence my post.
It seems like I am spending a lot of time on post-hoc justification for what should be a self-explanatory post (post 10).