So it turns out that Plexamp is a little buggy, and installing it from the AUR is not guaranteed to work. Instead, you can install it via Flatpak.
flatpak install flathub com.plexamp.Plexamp
Once that is installed you can run it via
flatpak run com.plexamp.Plexamp
However, you may not always want to have to type it all out on a terminal, and it is not found by default on Rofi, for example. So Flatpak applications are usually found in /var/lib/flatpak/app/. Moreover, at the time of this writing, Plexamp can be found here:
/var/lib/flatpak/app/com.plexamp.Plexamp/
Its .desktop file can be therefore found here
/var/lib/flatpak/app/com.plexamp.Plexamp/x86_64/stable/active/export/share/applications/com.plexamp.Plexamp.desktop
Therefore, what you can then do is:
cp /var/lib/flatpak/app/com.plexamp.Plexamp/x86_64/stable/active/export/share/applications/com.plexamp.Plexamp.desktop ~/.local/share/applications
And after you copy it there, you’ll start to find it via Rofi