Did some testing last night. Managed to fix it while launching from Steam (added apps to Steam), I’ve set “Web Browser” controller template (will customize template later if needed).

Just tested launching from App Menu, looks OK.

I think Steam needs to run, even in Desktop Mode for controller to be fully functional. So restarting Steam App helps.

Not sure if running an app triggers “Gamepad” settings for controller even if it is launched by App Menu.

Btw, didn’t had to do that for “Free Tube” flatpak, controls were working, but sometimes would just stop. I installed other apps later, then I noticed the issue. That is why I didn’t suspect on Steam App at first place. At that time I didn’t put Desktop apps in Steam Library so maybe that is necessary too.

Anyway, it works now.