Where can I find the command(s) that are executed when launching an application from the taskbar?

[Aravisian]

@gwiesenekker

but does anyone know what command is being executed when you click on a shortcut in the taskbar? Perhaps the corresponding .desktop shortcut is executed, but what command tries to execute that shortcut?

The command is the launch command. For example, if you are launching Gedit, your launcher command could be gedit %U

A panel shortcut is not a separate instance to be launched, just a redirect to the launch information for that .desktop file.

For your initial problem:
Have you yet tried:

sudo apt install appmenu-gtk2-module

?