Yeah, you could do that. I'm not sure how good it is. It may just be pulling out left/right if it is PCM audio. If it is dolby digital, then you might have a problem.
All streaming devices nowadays will only have HDMI output (Apple TV, Amazon Fire, Roku). The only good streaming device that does have a true left/right output would be a Sony Bluray Player. However, they are definitely limited on streaming apps. I know they will support Netflix, Pandora, Hulu. But some of the more esoteric streaming may not be support in Sony Bluray software. It's my opinion that Sony Bluray is the best streaming platform as far as audio and video quality. However, it's not a "any streaming app" platform.