Or a cheaper option of a blue ray player (I have a six year old sony blue ray player which has the Amazon video app )which you can probably get now for under $50.
As opposed to the 'use the native UI on your phone, then hit one button to cast it' experience.
We are Amazon Prime subscribers, but I can't remember the last time we watched anything on Amazon Prime Instant Video... it's just too much of a hassle...
"Dad, can I watch a show?"
"Sure honey <give her the phone>"
Now?
"Dad, can I watch a show?"
"Sorry sweetie, but I'm feeding your brothers right now, maybe in a little bit..."