When we don't have a wallpaper cached via Periodic Background Sync API, we shouldn't show the button or disable it. I would prefer disabling it so we can at least see if Periodic Sync would be available.
I'm in the middle of restructuring this, but even with the current code there's always a fallback image in the cache in the case Periodic Background Sync hasn't run (successfully) yet.
When we don't have a wallpaper cached via Periodic Background Sync API, we shouldn't show the button or disable it. I would prefer disabling it so we can at least see if Periodic Sync would be available.