I'm running a very similar setup and am looking for a better way to stream games from my local gaming pc. The current steam in-home-streaming is too delayed in reaction for my liking and my nvidia shield doesn't really support the latest and greatest like av1 or pyrowave. Has anyone a good idea how i could achieve it in one machine? I don't really want to buy into a steam machine or something just to play games.
There's probably something with sunshine / moonshine /artimis or something but I'm not up to date on that cluster of things.
With Gigabit LAN, I'm using Sunshine on my PC and the Android Moonlight client on the TV.
For Input, I've a Logitech K400 via USB dongle on the TV and Xbox controllers via Bluetooth.
Is there a slight lag? Yes. Are the majority of games I want enjoy on the couch and play on the big TV playable? Absolutely. I wouldn't play CoD but many many games just work well even with a Little lag.
PS: can also recommend playnite, giving a unified tv/Controller experience with access to multiple game stores , local Games and emulator support.