I understood some while ago that the train operator is keen to operate additional trains and that it is Network Rail who are being intransigent, claiming that due to maintenance requirements they cannnot release the track.
I wonder where you got that information from as it seems to be rubbish.
The truth is that trains do run a lot earlier and a lot later but in the wrong direction. While there is no train service serving Southend from London that would allow passengers to catch the first easyjet departures of the day the first train from the airport heading to London leaves at 04:35 followed by a 5:05 departure.
At night there are London departures right up until 00:18 which arrives at Southend at 01:20. From the airport the 23:05 is the last London bound train of the day which is of no use if you are arriving at anytime after 22:30.
Stobart doesn’t help matters by forcing late arriving passengers to leave the terminal who would be prepared to wait for the 04:35 departure to London.
All of London’s airports with much better 24/7 public transport links still have passengers that spend the night in the terminal so they don’t have to travel in the early hours but that is not an option here at Southend.