ETD is the date and time which an aircraft is expected to depart from an airport. It is also called expected time of departure or Estimated Off-Blocks Time (EOBT) in the 'airside world'.
Every flight has a Scheduled Time of Departure (STD), the date and time when an aircraft is planned to depart. All aircraft related processes (loading, catering, pax transport, boarding, push-back, ...) depend on that STD and plan their resources and equipment accordingly. Suppose there is a delay and no ETD is set. All related processes dispatch their resources wrongly. The STD time passes by and the push-back tractor is still waiting.
Who should set an ETD and when? In my opinion it should be the ground handler responsible for the flight. They should set an ETD as soon as they become aware of a potential delay. Which, in fact, could already be when an aircraft is arriving too late.
Obviously you can differentiate between a Staff ETD and a Public ETD to communicate expected delays more frequently to staff.