In case you're confused, eterm was just proposing this as an option. At this time the Twitch app does support casting from the desktop, Android, and iOS [0]. This roundabout solution would only be needed if Amazon decided to remove that functionality.
You don't have to dig around for the underlying video stream, you can actually just paste a twitch stream link straight into the vlc network stream dialog, I'm not sure what magic it does but that's worked even well before this new version.