But what does it mean if that box displays "No Current Shift" instead of your expected promise times?
This simply means that you are not currently within any defined daypart hours. Promise times are defined to default by daypart - breakfast, lunch, dinner, etc. -- so if you aren't in a daypart, you have no promise times.
This can occur if you are looking at the screen after hours, or if you've defined your dayparts to have gaps in the time. If lunch ends at 3:30 and dinner starts at 4:00, you've got a gap.
To check this, navigate to Manager Home > Configuration > General > Store Setup. The daypart hours are in the middle bottom. Check each one to make sure no gaps exist.
Read more about defining your daypart hours.