Earlier quoted context omitted.
Apple TV lets you watch all three (iTunes, Netflix, and Amazon Prime). Airplay is a thing.
Ah, but I want to be able to view Amazon Prime without resorting to an Airplay workaround. For my living room use case Airplay isn't an option; I'm looking for an integrated unit.
How is that a "workaround"?
An iPad, iPad Mini, iPod, or iPhone is your remote - the AppleTV remote is horrible anyways and I've personally long ago tossed it. Fire up your new remote and play anything you like... it's incredibly simple.