Re: What console?
Just to throw something else into the mix - I just got a nVidia Shield Android TV, and have been impressed with how streaming from my nVidia GTX970 equipped PC works. For reference, my PC is basically 4 years old, bar the 970 which arrived earlier this year, and has a gigabit cable connection to the Shield.
I've played a bit of Grid2 and some Fifa 16 and found it certainly meets my needs, and gives me flexibility to play on my PC in my cave, (and you can use the Shield controller on a PC in wired mode) or on the main TV with the kids. I was concerned about the latency between controller and game, but both my (adult) test subjects said it didn't seem to be an issue, and I can't say I've noticed it being an issue myself.
Of course, it won't help if you want console only games, but I thought I'd throw the suggestion out there....