Out-of-the-box, after clicking the Play button, a window with title "Elite -Dangerous (CLIENT)" pops up and instantly closes again. Turns out that Steam launches the game with the shell environment variable PROTON_USE_WINED3D11=1 which disables DXVK and this seems to be the source of this problem on my system. I use a shell script wrapper now which un-sets PROTON_USE_WINED3D11 and this makes everything work perfectly for me.