Hi Jeff
There is a permission mark staff course as complete totara/program:markstaffcoursecomplete that is allowed for the staff manager role by default. This should allow a manager to mark a course as complete via the program.
Permissions are applied to all users with that role so that means if a permission to mark a activity/course complete is provided then the user would be able to mark all users complete - not just their staff . This is a limitation of the capability system
Regards