Here's what you should have done: get a TV or a dongle with DLNA support. Then your PC will show up as a source and you can play any goddamn thing you want. Or you could have got a TV or a dongle with android on it, then you could have just downloaded Amazon's app.
[0]no "official" way at least
What? Pretty much every other video app/player I can think of works with chromecast.
http://www.androidpolice.com/2017/11/14/amazon-music-app-add...
Netflix works on just about every device you can watch video on. In the living room, that includes everything from the latest top-end game console (e.g. Xbox One X) all the way to inexpensive streaming sticks, cable boxes, etc.
Some streaming services work on a limited set of devices (partly because they likely have to re-implement their app for each platform). Amazon Prime, Hulu, and HBO Now are examples.