First of all, corporations like airlines, plan for medical emergencies and the death of passengers. It's gonna happen statistically based on how many people who fly everyday. Second, they are supposed to train their employees on how to deal with the situation appropriately. Dealing with a medical emergency could include, stopping the plane to take the lady to a hospital, working equipment, and someone on the plane who can use the equipment. When working for Luftansa one summer a plane was late arriving in Dulles because they stopped the plane, on their way out of Europe, to take someone who was having a heart attack to a hospital. Airlines are late all the time for non-medical emergencies and people are getting used to the delays and are smart enough to understand the situation. A good manager would deal with the situation and concentrate on maintaining their reputation for quality service.
When in flight the airline industry doesn't want to stop the plane anywhere but their final destination. The main problem is how much stopping over would effect revenue as it uses up more gasoline. To bolster profits, airplanes have been carrying less gasoline in the last few years, in order to cut down on costs, and they would probably have to refuel at the stopover point. This would be very inconvenient for the airline as being late effects its flight statistics. Airplanes are very expensive to operate and they make a lot of money each flight so any sort of lawsuit awarded might begin to look paltry in regards to the total revenue and operating costs of a routine flight. So the costs for dealing with an emergency could outweigh the damage to their reputation.
Lawsuits are also capped for pain and suffering, so the death of a normal person would be worth less then a wealthy person and the lady is a racial minority, most likely in a coach class flight, so her life could be expected to be worth less to the airline then a business passenger. This is not inheriently due to overt racism or sexism but to the inherent racism and sexism of statistics, as racial minorities and women are on average expected to make less money then white males and so could expect less of a payout based solely on income (the uncapped portion of the lawsuit)- which means that corporations ignore established economic protections more frequently for these groups of people as its cheaper. While these were not high level employees who made the decisions, airlines communicate to their staff which passengers are most important to keeping their jobs.
In many ways you get what you pay for, but the airline industry is more monopolistic then competitive, so the lowest bidder is able to ask for almost as much money as high class airlines who take better care of their passengers. So in the airline industry, people who are trying to save a buck, are unlikely to care about minor statistical problems with their reputation, as the death of one passenger matters more to her family then the millions of people who fly every year, particularly those who haven't heard the details of the case. On the other hand people who find airlines to be late more often are much more likely to expect that to be a flight problem so pay more attention to flight statistics regarding timeliness. The airlines are counting on this indifference, and ignorance of specifics as passenger tend to have more preference for noting statics that are likely to affect them.
Through labor union politics and corporate subsidy entitlement (as well as the politics regarding economic development and theft in third world nations), we have been propping up the idealogy of, "too big to fail," which means as an employer of many workers and the transportation of many consumers, the airline industry can basically claim its continued viability as a business or stock profits are more important then the death of one passenger. (For the sake of total statistics it is easy for industry lawyers to ignore the detail that the vast majority of people who fly airlines are relatively healthy and unlikely to fall over dead in a couple hours flight.) Lawsuits are designed to discourage this type of behavior but instead of being too high, money recouped from lawsuits might be too low as its cheaper to allow passengers to die then to adequately prepare for emergencies or to interupt their routine.
American Airlines should not be allowed to claim exemption as they do not need to work with very sick people as their primary function unlike doctors and hospitals, however they will likely continue to make the claim that they are 'too big to fail' as they don't want to give up any of their revenues to the family of the diseased woman; a payout would affect their stock prices whereas one lawsuit won't put the airlines out of business. Right now they are in a bargaining phase to limit the damage to their bottom line more then fighting to stay solvent as a company.
The upper management in this industry has been feeling quite cocky lately. In recent news the 'struggling airlines,' much like the oil companies who were subsidied to help pay for losses incured by Katrina, cut employee pay, benefits, and hours, to give their CEO's bonuses which pissed off their unions. It's also a bit rude to go tell the investors how much you ripped off the federal government by convincing the american people they needed to put their taxes towards keeping the airline industry afloat. If it wasn't profitable they wouldn't be flying planes.