I use this exact setup, heavily. I’ve used both XBox controlllers and Switch Pro Controllers (the pros unfortunately have a weird driver issue, where you don’t get full analog range out of the sticks). The controller connects directly to the Shield via Bluetooth, and the Steam Link app recognizes them, no problem.
The Steam Link app is occasionally quirky, but overall, I’m very happy with the setup. Have played Elden Ring, Hollow Knight, Celeste, Armored Core 6, Mass Effect, all this way. My wife has done a bunch as well. We’ve even done multiplayer with Tales of Symphonia and Plate Up, and I’ve done emulated games with Dolphin, although that was a fair bit trickier to setup.