I don't see where this has been posted by others (my apologies if I've missed it)... but I sent a written complaint to Amazon regarding the Apple TV. I'll paste the response below.
I'm not exactly sure how to take it, because it seems like they're saying "we're working on it" - but there is also no overt claim that an Apple TV app is in the works. So my advice would be to keep logging written complaints with them until we get it.
Here's the email from them:
__________________________________
Hello Derek,
I understand from your email that you want to use Amazon video App on your Apple TV.
Currently Amazon Video App are is not compatible with Apple TV, I can imagine this may be extremely inconvenient to you. That's definitely not we want our customers to experience, and I'm truly sorry.
Please note that the availability of Amazon Services is completely based on our agreements with the device manufacturers, please be assured that we are trying our best.
However, this does not mean your inconvenience and disappointment will go unnoticed.
We have recently received numerous requests of watching Amazon Videos on Apple TV. Please be assured that we're regularly working on improvements and adding more devices compatible to stream Prime videos.
You have been our loyal customer from about 13 years and the last thing we want is to lose you. Because your satisfaction is something we work hard to achieve.
To ensure utmost attention, I've considered your comments as feedback and personally forwarded them to the Amazon development team.
We are working very hard for the betterment of our products so that we provide a service that works best for our customers. I’m truly sorry for the trouble caused in this case.
We do value the feedback of our customers and make every attempt to integrate their comments or suggestions into our stores and procedures.
We have team of experts who takes into consideration each and every feedback of our customers and brings them into practice.
It might take some time, but they'll definitely do their best.Thank you for letting us know what you want from your Amazon Video service.
I do realize that the Airplay feature has it's limits and it's not the same without an official app.
As always, please feel free to contact us should you have future suggestions, questions or comments.
www.amazon.com/gp/help/contact-us/digital-video.html
I hope that you will honor us with another opportunity, to prove the quality of our service to you in the future as we look forward to seeing you again soon.
We'd appreciate your feedback. Please use the links below to tell us about your experience today.
Best regards,
Suhel N