I don't think that's the case because Prime Video is on Xbox and Playstation, where they instruct you to go to amazon.com to buy/rent videos. It could work exactly the same way on Apple TV. So why are they not on it?
Presumably just like the Prime Video app on iPad & iPhone: you must buy from the website, because buying thru the app requires a nontrivial percentage of revenue (on iOS, 30%) go to the platform provider.
While that's true, you can get around it by just not having a store in your app at all. This is how Amazon's existing iOS Kindle app -- and Instant Video streaming app -- work, and (barring politics on Apple and Google's parts) there's absolutely no reason that they can't make apps for Apple TV and Chromecast that work the same way.
Apple's refusal to even allow links to external stores is petty and obnoxious, but Amazon is essentially playing the same game: if you won't let us play the way we want, we're going to make life difficult for our own customers while clutching our pearls and telling everyone that you made us do it.
It must be another reason... But what?