I don’t have a contract with the airport. Easyjet does. Your car traffic metaphor is not accurate as those are entirely unrelated services. You could have walked, taken other roads, flown, taken a taxi, etc. to get to the airport on time. But there is no way to get to your flight without being processed at the airport.
This is the same logic as contacting UPS because your package is late. But you shouldn’t do that, call the person who sent you the package. Their contract is with them, and you are not their customer.
I’m not a customer of the airport, I’m a customer of the airline who uses the airport to fulfil part of their agreement. Their supplier failed, and thus the airline needs to provide compensation.
I don’t think the airline should be on the hook ultimately. They should have some kind of SLA with the airport that says if their passengers get to the airport within the allotted time they will process them or pay X as compensation, and they should activate that. But how is a customer going to go after an airport in a foreign country that they didn’t pay anything to? It makes no sense.