Here is what I wound up doing to work around this issue:
Launched InDesign CC 2015 (version 126.96.36.199) with Site Basic as the workspace specified to show at launch.
Clicked on a different tab on one of the palettes and quit InDesign, thereby creating Site Basic_CurrentWorkspace.
Launched InDesign again, which showed the Site Basic_CurrentWorkspace configuration. Clicked on the tab that originally was selected and quit InDesign again. This resulted in Site Basic and Site Basic_CurrentWorkspace now being identical.
Copied Site Basic as Essentials and placed it in C:\Program Files\Adobe\Adobe InDesign CC2015\Presets\InDesign Workspaces\en_us, replacing the version of Essentials already there.
Launched InDesign again and opened an existing document, which immediately changed the workspace to Essentials. Clicked on a different tab on one of the palettes and quit InDesign, thereby creating Essentials_CurrentWorkspace.
Launched InDesign again, which showed the Essentials_CurrentWorkspace configuration. Clicked on the tab that originally was selected and quit InDesign again. This results in Essentials and Essentials_CurrentWorkspace being identical to each other AND to Site Basic AND to Site Basic_Current Configuration.
Copied these to a location on a file server.
Wrote a .cmd file that copies all 4 of these workspace files to the correct location for the user, then launches InDesign CC 2015.
End result: When a user launches InDesign CC 2015 via this command file, the Site Basic workspace is shown. When a user creates a new document, Site Basic still is shown. When a user opens an existing document, the Essentials workspace is shown instead, but since it is identical to the Site Basic workspace, the user has all the same palettes and other workspace configuration as desired.