I confirmed that no one on our admin team made any changes recently that would have caused this, so I'm assuming it was a forced security enhancement. I also tested and confirmed that even if a user is given modify all on the Contacts Object AND has the "Manage Customer Users" enabled, they are still unable to access the View/Enable Self-Service pages.
The only workaround I've found is to give the user the "Manage Self-Service Portal" permission, but then that also gives them access to the Setup & Configuration Menus AND access to modify global Self-Service Portal changes.
We are working to migrate to Communities later this year, but I need a way to get this back to the way it was till then.