And the reason they used to give is that "Chromecast doesn't support Amazon Prime Video and that's confusing for users" - well guess whose decision that was?
Same thing with the Apple TV. And they apparently said that Apple wouldn't let their app into the store which is BS, since they already have an app for Prime Video for iOS.
While it doesn't support streaming the video directly to a Chromecast, if you don't mind burning your battery you can stream the whole screen of the device to the Chromecast.
You cannot any more. There is some kind of piracy protection that prevents it. I tried streaming the marvelous Mrs maisel to Chromecast via fullscreen recently and you can't see any video when you do, only audio.