Singing in for the iTunes Store is separate from signing in for iCloud.
iCloud has recently changed, and I think that may be a reason for legacy devices [Apple TV (3rd generation or earlier)] to lose some iCloud functionality, e.g. when those users have other devices that have been upgraded for iCloud.
There appears to be no official “discontinued” statement for this functionality and this model device, but quite a few users reported the same. Contact Apple Support for an official response, if they’re giving any.
No new software updates are expected for the 3rd generation model, as it is labeled obsolete for support purposes. So this issue won’t be fixed in that manner.