Smilin Ed,
To accept arriving and/or departing public transport flights, in Greece just like the US of A, an airport has to be fully open and manned. That means having a fire service ready for action at the Category needed for any aircraft that lands or takes off, ATC open and operating (Approach and Aerodrome, perhaps combined), lighting operative, Navaids etc etc.
It's not like an aircraft carrier, where these things are usually available because no-one can go home, there's enough people to work 3 shifts, and Uncle Sam picks up the bill without needing to make a return on capital.
So, believe it or not, there are airports that do close up when they are not needed for commercial operations, like at night. No-one pays them to stay open 24/7, so not only do they not do that, they couldn't do that. And in spite of my attempt at humour above, I think you'll find some in the USA.
An FBO, incidentally, is a business operating on an airfield, not necessarily an airfield. But some FBO's do own and operate the airfield as well, so confusion is forgiveable.