1) Reversion to the mean. This is what the above person believes and that this likely won't happen again.
2) Indication of a trend. The pilot is actually incompetent, and this will happen more frequently with this pilot than an average pilot.
Regression to the mean is simply that any given datapoint is most likely to be the mean, or close to it. This means that any exceptional data point,up or down, can be expected to be followed up by the mean.
The example of this being misinterpreted that I am familiar with is that of a flight instructor's belief on training. When a pilot performed well, they wouldnt comment. If a pilot performed poorly, they would be punished. They believed this was better because when they praised a pilot, they would usually do worse the following run, and when they punish them they do better. This isn't technically wrong, they are just ascribing causation where there is none. I think I saw this example in Signal vs. Noise, but I'm not sure.
Basically, regression to the mean isn't a reason to pick someone who did poorly, it's a reason that that person will do no worse or better than they do normally.
Flip 100 coins. Take the ones that 'failed' (landed tails) and scold them. Flip them again. Half improved! Praise the ones that got heads the first time. Flip them again. Half got worse :(
Clearly, scolding is more effective than praising.
While, of course, human performance is not always equal in the same way our coins are, the fact that performance (or whatever it is you are measuring) will still regress towards the mean still holds true. The coin example just gives an extremely obvious demonstration about what is happening when things regress towards the mean.
https://www.google.com/#&q=%22how+to+win+friends+and+influen...
Humans aren't perfect improvement machines, but they surely beat a random variable?!
Neither the GP's quote, nor the OP, are making the statement "I understand that this mistake was an outlier, you'll probably never make this mistake again if you remain unchanged".
The claim being made is that acknowledging the mistake and learning from it can dramatically reduce the base probability of such mistakes happening again.
The question is: is the day-to-day random variation much bigger than the day-to-day average increase? If so, regression towards the mean makes total sense.
This is psychological insight, not statistical insight.
3) Out of remorse, pride, being shocked out of complacency or some combination of these, the man would do his absolute best the next day, especially for the very same pilot.
If labour is so abundant that there's little problem in replacing any given worker, people can and will be fired for trivial offences. If there's a shortage, or there's a considerable on-boarding process, less so.
The threat of a firing-at-any-moment also makes for a more tractable workforce, at least from a management perspective.
Not everything is a financial metaphor.
I think more appropriate in this context is learning though. And I would suggest here multi-armed bandits (with every arm representing a pilot).
https://en.wikipedia.org/wiki/Multi-armed_bandit
There is tradeoff between exploration and exploitation that has to be made.
The goal in this case corresponds to a particular type of bandit. It is to postpone death for as long as possible by pulling the right arm. I actually didn't find this type yet (a mortal multi-armed bandit has a birth-death process of the arms themselves).
Edit: This is only about the learning process on the non-pilot side, as one of the other commentators already articulated.