Yes, having different versions of joplin.desktop or otherwise installing different icons based on distribution would be one workaround. It would probably be best both approaches were accounted for: the install script choosing a different icon based on distro and distros with relatively unique icon themes specifying a preferred icon for Joplin.
See this line here or this section here where the install script could be adapted to account for different default icon themes. (The install script already checks for the distro in this line here, so it shouldn't be terribly difficult for it to check for the distro when downloading the icon.)
It may be useful to check if you're even using the default icon theme on Linux Mint, though. This tutorial discusses changing the icon theme, but you can also use it to see what theme is currently selected:
As it is, the icons in the Joplin repository seem extremely disorganized, so it's hard to tell what's currently in use and what's not. (I feel like a README.md in the Assets folder could help in this regard.)
And there are yet more instances where things are slightly "off", like the high-resolution favicon for this forum having hardcoded corner radii when both Safari and Firefox use manual clipping. So you get things like this (from the Firefox "New Tab" screen):
Or this (from the Safari "New Tab" screen):
And the incongruous color icon in the GNOME tray (when Joplin does, in fact, include a monochrome icon that it just doesn't use here):
While it does use a monochrome icon in the macOS tray (which appears to be this icon? it's not clear):
So, yes, there is a lot of potential finessing to be done with all of the different use cases of the icons included with Joplin.
