Figured out why Amazon was in no rush to put the Prime Video app on AppleTV for the longest time: while all the Prime Video free streaming stuff is readily available, anything to rent or buy must first be purchased through a regular web browser. That's /arguably/ easy on mobile devices, as you can just fire up the browser, but AppleTV doesn't have a browser on the device.
It threw me for a loop, because I'd been using the Amazon Prime Video app on PS3/PS4 for a year, and it allows users to just clickthrough a purchase to their registered Amazon account.
Apple's policy to prevent people from entering their CC information in apps and then being held accountable was understandable, but the idea that I have already got an account on a server and can't access it is utter foolishness.