> I will click the green "Play" button, it will change to a blue "Stop" button, as if the application was running, then shortly after silently switches back to the green Play button again, without any visible error and without actually starting the game.

You might want to enable Proton logging and have a look at what it says is going on.

https://github.com/ValveSoftware/Proton/?tab=readme-ov-file#...