I fully understand airport and airline operators pushing for this as they are experiencing massive damage to their businesses as we are, but isn't the concept of airport testing fundamentally flawed because of the biology of the disease?
As I understand it, according the the WHO there is an incubation period which averages 8 days from exposure to symptoms. However incubation can be as short as 2 or 3 days, but in 10% of people is 14 days or longer. We only test positive when we become infectious around 1-2 days before symptoms start. Hence, and most importantly, we test negative for the majority of the incubation period. Asymptomatic carriers also have the same variable incubation period until they too are infectious, although they may never know it. We stop being infectious around 10 days later. Note the distinction between infected (we have it in us) and infectious (we can give it to others).
So as an example, and if the Donald Trump timelines are to be believed, he was infected in the Rose Garden ceremony Sunday, tested negative Monday, Tuesday and Wednesday (assuming he was tested every day, which hasn't been confirmed), tested positive Thursday and was taken ill Friday, i.e. he was on the shorter side of the range observed.
Assuming the tests are accurate (and as others have pointed out, much of the existing testing isn't perfect) a negative test only says we were not infectious on the day of the test. A test at the airport immediately before departure may give confidence that you are safe to fly, but it says nothing about whether you are incubating the disease and will go on to become infectious in the next week or two. Nor does a negative test at the arrival airport.