There are still a few more things to configure, such as getting RetroPie's shortcuts to work in Dolphin, but at least the games start!

If you are referring to the Hotkeys, you would have to be running lr-dolphin for that to work. The standalone emulator does have the ability to configure hotkeys, but you will need to do it through the dolphin settings menu.

The issue was not with the snap or with Emulation Station! dolphin-emu works dolphin-emu-nogui does not. That's it. I don't know why and I no longer care. I changed to configuration files to make sure that the nogui version is never used. That did it. There are still a few more things to configure, such as getting RetroPie's shortcuts to work in Dolphin, but at least the games start!

@mitu Actually, the first parameter is -e. The ROM file's path is the second parameter.

You're right.
Then I don't know why it's not working - there's nothing special added by runcommand , it just runs your script with the parameters given.

Originally, I forgot the double quotation marks and I got an error if the ROM's filename contained spaces, so I know this is correct.

I could actually combine these two by having instead:

snap run dolphin-emulator.$0 "$@"

Maybe with a hard link between the two, but that feels like overkill. I tried the version with -e and the ROM file's path, and that worked from the command line.

snap run dolphin-emulator.dolphin-emu-nogui -e "$1"

since the 'rom' file is actually the 1st parameter of the invoking script, 2nd one is empty. You can put the script anywhere, it doesn't have to be located under /opt/retropie.

I am not running Wayland. xset actually works outside of the snap.

It seems the integration of snaps is not that seamless. The snap may have it's own $PATH and can't find the system's xset binary.

emulators.cfg for the Game Cube:

dolphin = "/opt/retropie/emulators/dolphin/bin/dolphin-emu-nogui -e %ROM%"
default = "dolphin"
dolphin-gui = "/opt/retropie/emulators/dolphin/bin/dolphin-emu -b -e %ROM%"

Contents of /opt/retropie/emulators/dolphin/bin/dolphin-emu:

#/bin/bash
snap run dolphin-emulator.dolphin-emu -e "$2"

Contents of /opt/retropie/emulators/dolphin/bin/dolphin-emu-nogui:

#/bin/bash
snap run dolphin-emulator.dolphin-emu-nogui -e "$2"

Contents of: /dev/shm/runcommand.log:

Parameters: 
Executing: /opt/retropie/emulators/dolphin/bin/dolphin-emu-nogui -e "/home/eje211/RetroPie/roms/gc/Wind Waker.iso"
/snap/dolphin-emulator/118/usr/bin/xdg-screensaver: 629: /snap/dolphin-emulator/118/usr/bin/xdg-screensaver: xset: not found
libGL error: MESA-LOADER: failed to open i965 (search paths /usr/lib/x86_64-linux-gnu/dri:\$${ORIGIN}/dri:/usr/lib/dri)
libGL error: failed to load driver: i965
libGL error: MESA-LOADER: failed to open i965 (search paths /usr/lib/x86_64-linux-gnu/dri:\$${ORIGIN}/dri:/usr/lib/dri)
libGL error: failed to load driver: i965
libGL error: MESA-LOADER: failed to open swrast (search paths /usr/lib/x86_64-linux-gnu/dri:\$${ORIGIN}/dri:/usr/lib/dri)
libGL error: failed to load driver: swrast
Failed to initialize video backend!
/opt/retropie/emulators/dolphin/bin/dolphin-emu-nogui: line 2: 66538 Segmentation fault      snap run dolphin-emulator.dolphin-emu-nogui -e "$2"

I am not running Wayland. xset actually works outside of the snap.

The command that RetroPie uses to start Dolphin is (from memory; I'm not by the computer): snap run dolphin-emulator.dolphin-emu-nogui -e "$2"from a script that's called dolphin-emu-nogui.

What could be different between running Dolphin from Emulation Station and from the GUI?

What's the command that you configured for runcommand to start the emulator ?