Amazon's fork of Android is not a platform Google is obligated to support. Amazon chooses not to include Chromecast support in their Amazon Video Android app
It should also be noted that Amazon's fork of Android, with the exception of the Linux Kernel, is closed source. They're not even a very good open source citizen as I don't think they've contributed anything back to AOSP.
That's an unfair statement in the great scheme of open source. When you fork the Android Open Source Project you only get a backbone version of the OS. You start at level 0. Google keeps all its custom proprietary and key technologies closed (rightfully).
Amazon is keeping its customization closed as well.
As far as contributing back to the AOSP, that's another area where Google's iron grip on Android shows.
You can report bugs and help to patch things, but there's no actual way to contribute anything meaningful to the roadmap.
What you fail to realize is that a lot of Android OEM's contribute code back to AOSP. That is being a good open source citizen. Amazon has yet to contribute anything.
