Why do airliners have to file delays through their handlers? Often when listening to ATC (commonly Dublin), when an ATP requests push and start, the controller will reply, "You have missed your pushback window", and/or ask the pilot to call their company or handler to "file a delay" or "update your flight plan with your off-blocks time".

I have two questions about this but I expect the same answer answers both.


Why is it necessary to re-file the plan, rather than simply activating the flight plan with the actual departure time?
Why is it necessary for the pilot to change to the airline's frequency, chat them up, change back, and then wait for the electronic flight plan to be updated, rather than talking to the controller directly?
Since you mention Dublin, I am going to give an answer that describes the situation in Europe. Other parts of the world may have different procedures, although I believe they will be similar. 

Usually a flight plan must be activated within a given time window to ensure that there is enough capacity (airspace, ATC staffing, departure &amp; arrival airport constraints) along the route to handle the flight. If the flight is not departing with this 'slot' a new flight plan needs to be requested and approved by the Network Manager Operations Centre (NMOC) (Previously called Central Flow Management Unit (CFMU)). 

The controller is not responsible for filing flight plans and cannot be tasked with doing the paperwork for the crew. The crew therefore needs to contact their handler or flight operation centre to refile a flight plan. 

The NMOC will reassess the flight plan and verify that there is capacity along the planned route to handle the flight. If there isn't, the flight may need to be planned along another route or an additional slot delay will be given.