I'm sorry to report the solution offered by the Roku employee above does not work in fixing this bug in Amazon Music. Worth a try, but not the solution.
However, here is how Roku can fix this problem. Contact Amazon and say, "We're sorry, but your app does not work on our platform so we're removing it until you make it function properly."
But this is not going to happen. So long as Roku, Amazon and our ISPs can all blame shift by pointing the finger at each other, that's what they'll stick with, and paying customers will be left holding the bag as usual.
If a Roku employee replies to this message they'll say something like "oh dear, there's nothing we can do" but of course there is something Roku management can do. They can kick Amazon out of Roku until such issues are fixed. That would work, but it won't happen.
This site is broken too. Who shall we blame shift that on to?