what you described is the problem and responsibility of the individual. i don't see why it should be at the departure airport, that is an unnecessary scaling issue for countries/airlines.

make it at the arrival airport, if the traveler doesn't have a valid recent test, he can take it again at the arrival airport + mandatory quarantine (and by that i mean in a nearby facility escorted by police). If you want to travel, it should be your responsibility to not infect everyone else...