apple tv

If you really have a 1st gen then it no longer has access to the iTunes Store.

If its it’s a small black box then it’s a 2nd Gen and it can still access the store, you need to enter the password and code together. However to avoid exceeding the time limit it’s best to enter the password, then request the code and add it to the password

Get a verification code and sign in with two-factor authentication - Apple Support

Did you add the code onto the end of the password as one entry? Example: password1234

Not password -submit, code -submit . Don’t click submit until you’ve put it all as one

Once you've got the code, enter the password and the code immediately after it in the password field, e.g. password12345.

If you are attempting to input both together you can easily exceed the time limit. As above, it’s best to enter the password, then request the code and then add it to the password.