I recommend you get a game console - all three current and last generation consoles support Amazon video (Wii/Wii U/PS3/PS4/Xbox 360/Xbox One). Plus they can play games.
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..."