This is a very odd choice on their part.
Which unsurprisingly, doesn't support media from the Google Play Store. The only acceptable streaming device is one from a neutral third party (Roku, Tivo, etc.) who will support all providers.
Amazon makes some strange decisions.
By getting you to spend more time in their interface, they can add things. E.g. Prime Music had me spending a lot of time checking out playlists, some of which may very well lead to purchases of albums where only a couple of songs are available for free.
It also gets them penetration for whenever they add shopping to these devices, and e.g. lets you talk to Echo and ask it to bring up the top five best new phones on your TV for you to choose fro.
Getting Prime on Chromecast would help their penetration of Prime, but it might hinder their overall service penetration by giving some people a reason not to get a Fire Stick. Especially when the Fire Stick/Fire TV and Chromecast are all cheap enough that there's little stopping people from getting both...
It may very well backfire for them, but there's all kinds of reasons for them to want to push the Fire TV devices hard.
I don't have enough ports on my TV for a Fire Stick and a Chromecast (and everything else) so I'm going to choose the one with the most options (Chromecast).
They can lose a lot of users and still find it to be worth it for the extra engagement of the ones they did get.
Amazon really does not like Google. This is not a failure on Google's part, this is all of Amazon's own making in an attempt at product lock-in.
Is there a new, new way to get it to work that I don't know about?
1) Sideload Amazon app APK 2) Sideload Amazon Video app APK 3) Open Amazon app and log in 4) Open Amazon Video app
However, it wasn't a pleasant experience. The version of the Amazon Video app I was able to get working didn't scale well to large screens, didn't work well with remote-based navigation, and didn't seem to do HD. I watched one episode of something, and haven't used it again since.
In others: http://www.amazon.com/Amazon-com-Amazon-Video/dp/B00N28818A
For most users, installing the Amazon Appstore is required.
On the other hand, maybe Amazon is angry at Google for not making Google Play Movies/Music available on the 'Fire' line of tablets, and this is retribution?
The equivalent to that would be refusing to release Amazon Video on the Play Store, which Amazon has done.
Banning hardware products from the Amazon marketplace is an escalation.