I have the ShieldTV and play Skyrim on my HDTV.
I have run my own games from PC to TV but I have not tried the GeForce Now game streaming service.
You do need an nVidia video card I believe for the Shield.
I am wired and it works nicely so I am not sure if with wireless you would have the same issues as before.
Not sure what I can check for you that you can probably find from checking Youtube videos but I can try.
One thing to consider is grabbing some Powerline Ethernet equipment and hooking up that way.
https://www.amazon.com/Powerline-Computer-Network-Adapters/b/ref=dp_bc_5?ie=UTF8&node=1194444You will get a better connection vs wireless and that may fix your issues with the Steam device.
I use the ShieldTV mostly for video streaming, PLEX, Youtube etc and I love it.