How do airlines train their pilots to deal with the stress of delays?
(My answer is limited to carriers based in the USA, things can be very different in other regions, many countries pay monthly salary and culturally have different view of company expectation verses safety expectation)
Pilots are paid per hour of block time, they are getting paid for all that time waiting on the taxiway after leaving the gate and starting an engine. The flight plan will usually include an extra hour or two for flying to an alternate airport if the weather turns bad at the destination airport, this is not commonly used with modern forecasting but it is part of the plan. So generally a pilot should not have a strong financial motivation causing delay stress. If a poorly timed childrens party is causing distraction then my opinion is that the company should consider strongly encouraging this pilot find a less safety critical career.
Pilots in the USA in recent years do tend to get a lot of general stress-management guidance during training, even a fair bit at the private pilot level.(depending on the school) Often this training is lumped under labels like &quot;Human Factors&quot; or &quot;crew resource management&quot;, and FAA materials even include specific advice for avoiding situations just as described in your question they call it &quot;get-there-itis&quot;. The free FAA materials tend to be a bit superficial but better than nothing in calling attention to a possible problem, industry training courses on the other hand have some good depth. This type of training has been growing since the late 1960s, but it got a large boost in the early 1980s after the 1977 Tenerife disaster where a KLM captain apperently hot to make up for delays started takeoff without full clearance and slammed his 747 into a Pan-Am 747 taxiing in the fog, just at rotation speed, killing over 500 people.