Oh right. Guess I was still too preoccupied in my mind with coming from Windows, that I did not consider that option.
Still not much luck tho.
To figure out if any runners at all - if installed directly via Lutris - do work, I tested Mednafen. Specifically Pokémon Red/Blue and Crystal. Everything installed via the scripts on this website. Curiously, Red and Blue do work. Crystal does not.
So now the runner seems to work, but only with classic GB games.
With Crystal, the program just doesn’t start at all, just as with ZDOOM and libretro.
It says “Game is running” on the button for a moment, just to switch immediately to “Game has quit”.
Regarding the icons, I realized they do indeed show up after (quite) awhile. But only for some the icons actually appear. Heretic has the proper icon, all of the Pokémons do not.