"I wonder how airline managers would react when the AI system refuses to fly because it assesses the conditions as being sub optimal for profit or safety?"
Retrain it?
Seriously, if it made that assessment, it would prudent to find out why.
"The problem with neural networks is that their reasoning is not readily transparent."
That is certainly an issue. How do you certify it as safe?
I think, however, that this will eventually be solved. You can design systems to give information on their reasoning, and I suspect that will be required. In other words, the system would need to be able to describe and justify its reasoning, just like a trainee pilot.
I don't think that opaque strong AI (deep learning) systems will be accepted for executive reasoning without a huge amount of evidence that they are correct. They will be used very soon as part of sensor systems (at least for automobiles), as they offer perhaps the best hope for problems like understanding images.