Got it.
The issue with not ejecting an ExFat disk and having a problem because of that is not something I've run into in years of using ExFat disks to move media files between systems, but apparently it can happen.
Regarding the Mac mounting the SalkStream disk containing the music files, I'm not sure what Mac OS version you are using, but in the Finder there is a "Network" item in the "Go" menu that should list devices visible to the Mac on the network (see below). If you open Network you can just double click on an item and see if it has a volume (or more than one volume) that can be mounted on the Mac.
If the SalkStream is visible there but double clicking just gives you no result or "connection failed" then possibly there is something that can be done in the Linux OS on the SalkStream to make the volume mountable. That would be something Jim or someone familiar with that Linux installation could help with.
In principle the SalkStream is running Linux and is a device on the local network so it would be reasonable to be able to allow a local volume on the SalkStream to be mounted on a computer on the network. That's a very common thing, in general.