I have two-step authentication turned on on my Google account. My ATV won't take either my normal password OR an application-specific password generated from my Google accounts page. Does anyone have a similar setup and have you gotten it working?
Are you running an Apple developer iOS beta by chance? Because when I run iOS betas on my ATV YouTube never let's me log in until I run an official release of iOS.
If not try to restore your ATV and see if it works then.
Unless the ATV doesn't support google two step authentication, I used to use two step authentication a few years back and I had problems with certen devises/apps not letting me log in. So that could be what's stoping you, try turning that off and see if it works.
If none of that works, then I'm stumped and maybe somebody else can help.