Yeah, so something like steam link may use a nontraditional input that the flatpaks weren't setup to expect.
I've used moonlight myself quite a lot on Xbox with the controller, but admittedly a good game list like steam requires setup (or ES-DE or equivalents)
Might help your case if you set it up to launch steam on connect if it helps the controller input (again though not 100% sure it will, I just know moonlight shows its controllers as a Microsoft Xbox 360 so its pretty supported)