It's got nothing to do with the 'UK staying in the South Atlantic'. It has to do with a community for whom Saint is home and to them home cannot be anywhere else but Saint Helena. Building an airport was meant to give them more chance of self sustainability.
The Falkanders voted overwhelmingly to remain connected to the UK. Notwithstanding this fact I can't imagine any of them happy if we just turned up with a boat and said we are shutting them down.