TL:DR Emulators installed via Flatpak won't recognize steam inputs during remote play, However AppImages will work. So if you want to play PCSX2 via Steam Link from your main rig, install the AppImage version.
So this whole journey started when I wanted to play PS2 games from my PC on my Big Screen TV. Years ago I got a Steam Link, and outside of it forgetting my Bluetooth devices it's been quite reliable.
However when I switched over to Linux and installed my emulators via Flathub, I could start the emulator, but outside of mouse inputs, the emulator refused to recognize the inputs outside of those directly connected to the PC. I presume it's due to how Flatpaks work.
Reported the problem on the Steam Linux Beta github page, but that still left me without a solution. It's not like there exists a *.deb for every emulator, and if there existed one, it was out dated, so it was hit or miss if I could start a game with a controller.
However when I tried AppImages, it worked! I presume it's due to the fact that the AppImage runner is a locally installed package, which means it can hook into the systems inputs, like Steam Input. Thus far I've tested PCSX2, DuckStation and Retroarch, and everything appears to be working beautifully!
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)