Isn't there a very similar thing built in to tvOS 11? I think it's more of a syncing feature than a back up though, but perhaps it will work in the same as a back up when you get a new box - may only work if you have more than one box on your network?
I should just shut up until I read up on it again
EDIT
OK I looked it up
Home Screen Sync
This new feature will let you sync your apps and home screen across any Apple TV device connected to the same iCloud account - though both will need to be running tvOS 11 for it to work.