I am having the same issue with my gen 3 (but not my 4K, at least yet). I followed the Netflix guidance to sign in and out of my iTunes account. My device did not have 2 factor authentication.
Yesterday, I had a "chat" with a netflix person. They parroted the advice about signing in and out. And, parroted the advice to contact Apple for help with 2 factor authentication.
I had a live conversation with an Apple rep. They had me go through the steps to confirm the problem. They checked me out on 2 factor authentication including using the code with the password. Same issue. Their counsel was to contact Netflix again and talk to a senior adviser.
I recontacted Netflix again. After 20 minutes of the same parroting, she started asking me to supply the error code. It took me about 5 repeats of "their is no error code, that is not the problem, the problem is the constant signing out" for her to stop asking me for it. After conferring with the "Experts", she came back to tell me this is a known issue and it is affecting other subscribers, too. The "engineers are working on it and, in the meantime, please be patient."
Seems like there is nothing we can do on our end except complain. They messed up and the reps are parrots and nothing more.