With the guidance above I have managed to convince ArcGIS Desktop to use a common location for the Add-In. As our student machines do not have persistent User Profiles, the My Documents location is deleted on reboot. Students do not have a redirected or Roaming document storage location apart from OneDrive. Thus, in order for the Add-In to be discoverable by ArcGIS Desktop, I moved the Add-In to the C:\Users\Public\Documents\arcGIS folder and changed the ArcGIS XML file that points to the Add-Ins locations. This XML file can be found in c:\Users\%USERNAME%\AppData\Roaming\ESRI\Desktop10.6 and is called AddInFoldersList.xml. If I add the Public Documents path for the Add-In to this file and copy the Desktop10.6 folder structure to the Default user profile then all subsequent users will have the add-in available.
This may also be of assistance to those that are Virtualizing ArcGIS Desktop for their users.
From a silent install perspective it is a little trickier. You can either copy the extracted add-in and folder structure to the Default user profile on all your targets and amend the AddInFoldersList.xml or simply clobber what is in place. This is probably not something you would do for your staff if they already have added their own Add-Ins folder locations. With some Powershell scripting though, you can simply add the entry into the XML.
Hope that was helpful,