and then used a keyboard/mouse on the couch? (diablo 3 doesn't support controllers on PC now does it?)
Correct on the keyboard and mouse (not sure about controller support in Diable). I got a wireless Logitech mouse that I put the USB receiver into the Steam Link and then I routed a usb extender cable neatly around my living room and attached a USB keyboard to it (its a gaming couch keyboard that has built in mouse pad/tray extension). I just plug in the keyboard when I want to game on the couch and then put it in the closet when I am not.
Yup that. Personally I prefer this option over the steam link as the Raspberry Pi is more powerful hardware wise then the Steam Link. The CPU in the Steam Link is not powerful at all...I tried putting a gaming mouse on the Steam Link and the high DPI made the sound stutter...kid you not (known issue I found in some forums). Basically the CPU could not keep up the mouse refreshes and the sound suffered.
Also I should add...technically you do not need to plug the controller, keyboard, or mouse into the Steam Link to make it work. If the distance from where you are sitting and actual computer is short enough for good wireless signal...you can plug components directly into PC. My computer is too far away to route USB extenders or have good wireless strength.