Once upon a time (probably with RF8) I tried running RealFlight on my living room TV through my Steam Link hardware talking to my PC upstairs, but I couldn't get it working with my InterLink connected to the Steam Link's USB ports. Maybe it would be possible with the right setup, but at the time when I was attempting it and doing some research into the issues I remember having reason to think it was simply not going to work. RC-style controllers don't map all that well to generic gamepad inputs.
I vaguely remember somebody claiming a third-party utility might help with situations like this. I also remember thinking that even if I did get it all working, added latency (input in particular) might be a dealbreaker.
Streaming video is one thing. I suspect the larger obstacle will always be the controller connection. Depending on your setup there may be ways around that, though. For example, you could have a wireless dongle connected directly to your PC in another room and transmit to it wirelessly from the room with the TV.
If you do get anything working in a way that makes you happy, by all means tell us about how you did it!