@jonas - sorry for all the replies --
I figured out how to solve the issue. I vaguely remembered skimming past a comment somewhere that the flatpak version of Strawberry had some kind of issue (and I had downloaded the flatpak). So I uninstalled that, and got the 'true' version via the terminal.
This time, when I added my external drive to the directory, the file path actually looked normal... so that was a good sign. I rebooted the pc, ran Strawberry... and sure enough -- the library remained, and I was able to play songs as expected.
So, TL;DR for anyone stumbling onto this thread with the same issue I had - uninstall the Strawberry flatpak and get it direct from the terminal. I followed the steps on this link: https://linuxcapable.com/how-to-install-strawberry-music-player-on-linux-mint/
... and get the normal/stable version.