So I have a setup similar to this. I'm using steamlink hardware, with an Xbox controller dongle, with virtual here running. Virtual here passes the USB back to the computer making my PC think that it is directly connected. I have a link to bigbox setup in Steam, you have to make sure launchbox isn't running or bigbox will never open up through big picture. I can also launch xbox streaming and play xbox one games as well. Make sure you don't turn on controller support in steam, it makes the controller emulate keyboard and mouse functions and that will cause button mapping issues.
To take this to the next level I added Parsec to my gaming rig and I can stream games to any PC or cell phone as well as invite people to play multiplayer games over the internet. Online multiplayer with retro console games.