Earlier quoted context omitted.
cool, I didn't know Apple TV could be a Sunshine/Moonlight client!
It's perfect for gaming: silent, reliable network and Bluetooth, and supports basically any TV or projector. Works great with Xbox Elite controllers. Only downside is that keyboard/mouse can't connect to Apple TV, so it's controller-only.
For mouse, if you have an iOS device you can use the "Remote" widget on your phone as a mouse, and it works in the Moonlight client well enough to click through prompts etc.