I am using the GNOME edition.
The issue is the following.
After updating Proton, the SteamPlay titles don't start with native runtime, they have to be started with the Steam runtime first (after that they begin working with the native runtime).
[edit: It's odd. I tried another title, didn't launch at first with the native runtime. I tried launching Steam - with native runtime enabled - via the terminal to see what happens when I try to launch the game, but this time the game launched.]
Another issue is with the Steam controller. With Steam-runtime, games no longer detect it as a controller (it uses the desktop configuration). Meanwhile the controller works just fine with the native runtime enabled.
It's my first ever bug report, so I'm sorry if the style of it might be a little unorthodox so to say.