Are you using Wayland? If so, you need to launch Steam as steam -pipewire, and then select your display. Note that your screensaver will be inhibited while Steam is open with this option.

That being said, I would recommend going for a Sunshine/Moonlight setup if you can rather than using Steam remote play. I get much better performance with Sunshine and Moonlight.

Edit: I just read that steam games work fine for you. In that case, I'm not entirely sure why non-Steam games don't work.