At least for now, you’ll need to run moonlight through Steam. It uses Steam for input profiles, so initially it doesn’t seem like it will do much without steam running.

The original steam controller did have linux support without steam, but we don’t know yet how/when that will carry over to the new one. I also expect valve to eventually set it to work like the steam deck’s desktop controls, where it has default desktop inputs and you can hold start to swap to a generac xbox style controller inputs. However they haven’t announced that yet and I haven’t heard it mentioned in any reviews.