Apple TV asking for iTunes store account

That’s the same ATV model that I’ve got, which should be fine.

I suggest the following:

  1. Do a Reset (not Restart) from the System menu. I know you said it had been reset, but let’s be on the safe side. This will put it back to factory settings. It also doesn’t take long.

  2. Make sure your ATV is running the latest version of tvOS by trying to update it from the System menu.

Where did you set up the Apple account that you mention? If it still doesn’t work then it sounds like there’s something wrong with the account. You could try and set up a different one or ask Apple to help.