Amazon Video App disconnecting other devices from router.

The thing is, that the Amazon App is developed by Amazon themselves. Microsoft only offers the store to download the Amazon App from.

If other video streaming works on your Xbox (e.g. Netflix or whatever other streaming services you might have), and it's related to the Amazon app, you would have to contact Amazon support for that.

It appears that Amazon does have an app for the Xbox 360 according to the FAQs regarding this question.

https://support.xbox.com/en-US/xbox-360/apps/amazon-instant-video-setup

Install the Amazon Instant Video app

1. From Xbox Home, go to Apps, and then select Browse Apps.
2. Browse or search for Amazon.
3. Select Amazon to download the app. Once downloaded, the Amazon Instant Video app automatically starts.

To start Amazon Instant Video later, go to Video, My Video Apps, and then select Amazon.

That could be the issue, that your router does not have sufficient bandwith to withstand the connection for the console. Try disconnecting other devices from the router, then try and play and see if you get disconnected.

Hope this helps!

Do you get a specific error message? Or what exactly prevents you from connecting to Amazon Prime Video?

Are you able to watch Prime Video on other devices, or through your web browser?

$16? Sounds like you bought the movie, not rented it. If you bought it, it shows up in your account on all devices (phones, computer, Xbox console, etc). Rentals only on the device you rented it on. Hate to say it, but if I'm renting a video, I rent from
Amazon. It is usually a buck cheaper and there is no limitation on what device you watch it on---phone, tablet, pc, Xbox, Google TV, Tivo, etc.--anything with an Amazon Video app or a web browser.