The OP clearly stated that they were trying to sign in and the verification is what is required to authenticate prior to sign in so if you are facing the same issue then you wouldn’t be signed in.
I will reiterate what I posted above, not sure what it is that isn’t making sense.
When signing in you would enter the email associated with the ID, then you would be prompted to enter the password. This is where you would add the code, as it will not prompt to enter it separately on an older model and requires it be part of the password. However, there is a time limit so to avoid that it is required to manually request the code, refer to link posted above. This does in fact work and many users on older hardware have successfully used this method.
The 3rd Gen is over 7 years old, however Apple is still very much supporting it. This is evidenced by the fact that they continue to provide firmware updates and created a server version of the Tv app for that model. However, 2 factor authentication is optimized for newer software that the 3rd Gen hardware is unable to run. Therefore, it requires the aforementioned method.