How have you tried to use the MainStage sounds? I am using mainly the sounds from the Legacy section, the JamPacks and the and the Loops. When you download all sounds for MainStage and install them, they will appear automatically in GarageBands Sound Library. MainStage will install a lot of sounds. You may want to be selective, when downloading the MainStage sounds. They require a lot of storage.
Once you installed MainStage, go to the menu MainStage 3 > Sound library > Open Sound Library Manager.
You will now see a list of all available sounds. Open the "Legacy" section to find the Jam Packs. I would primarily install the Jam Packs.
After you downloaded the loops and instruments, go to the Loops panel in GarageBand and scroll down to "Reindex all Loops". Then restart the Mac to update the sound library. Now GarageBand should be seeing the additional instruments. You can use the JamPacks directly from the Library panel in GarageBand. Just click the Legacy section in the sound library, when you select the patch for a software instrument track. And the loops should appear in the Loop browser.
