3rd generation Apple TV can't connect to iCloud

Hello,

I recently had to change my Apple ID password. Today, my 3rd generation Apple TV could not connect to iCloud; I had to enter the new password. I received the 6-digit authorization code and entered it into the TV and received a message: "iCloud is not available.Try again later."

I repeated this several times, ensuring that the password and Apple ID were entered correctly. Same result.

I was on the phone with Apple Support for 45 minutes and nothing we tried worked. The Apple TV works normally otherwise. I restarted and then reset the Apple TV. I was able to sign into the iTunes Store with the same ID/password. I'm able to login to iCloud with the same ID/password from my Mac, my iPhone, and iPad on the same wifi network as the Apple TV.

I (and apparently Apple Support) am stumped.

Any ideas?

Thanks!