The Prisoner's Dilemma is frequently mentioned because it is one of the most fundamental building blocks of game theory. Almost everything else that we think of as game theory is either a variant of it, built on top of it, or is an attempt to tweak it.
The simplest version of the Prisoner's Dilemma succinctly explains both (1) why rational beings feel compelled to have a moral code in the first place, and (2) why they so often disobey the moral code that they themselves believe in. It's best for everyone to cooperate; but in the absence of a guarantee that everyone else will, it's in my best interest to defect.
Of course, you can't get from PD to police corruption in a single step. Lots of intermediate steps and computer modeling is needed to get there. But most of that is probably too technical for an Aeon article, so the article only mentions the first and last steps. A more detailed account of the models they used would be very interesting, but that's probably the job of actual scientific papers.
But the PD is faulty at it's core, because it doesn't take into account external factors. What if the criminals are in a gang that treats snitches very harshly? Or not in a gang but in a culture of 'omerta'? What if the criminals' career prospects are affected more harshly by a long vs short sentence? What if admitting guilt fast-tracks the criminal into a drug rehab course that the criminal is interested in? What if one police interrogator reads criminal A better and thus phrases the deal in a more attractive way? What if the criminal is on the third strike in a 'three strikes' location? What if the criminal is a devoted father, and can't possibly entertain the option of being away from his child for the term of the long sentence? What if the criminal is an institutionalised homeless person, and the idea of prison is attractive?
There are heaps of external factors that can affect this decision, from subtle to severe, and the PD doesn't model them well at all. As a base building block, it's not very solid.
All the complications you mentioned can be, and usually are, modeled as additional constraints on PD.
You can add arbitrary constraints on PD, such as a 50% chance that a third party will punish you for defecting. More importantly, you can play PD many times in a row and have each round's incentive structure depends on the result of previous rounds, sort of like encrypting in CBC mode.
The rewards and penalties don't need to be jail time, either. You can gamble with money, your life, or anything else you value. Usually it's done with some representation of money, because money is easy to measure and more intuitive to people who've never been in a prison.
The iterated (many rounds) variant is extremely powerful, as it allows researchers to simulate all sorts of complicated constraints. For example, other players might become more likely to defect on you if you defect on them three times in a row (three strikes). Certain players (the mob boss) might be much more interested in your performance than others, and defect on you much more severely when disappointed. You might be given an opportunity to reset your records (rehab or pardon) after a certain pattern of defection and cooperation, or maybe it will be game over (death sentence) after a different pattern. Iterate a few million times, and you've got a pretty damn accurate picture of how effective each policy will be in discouraging defection (crime).
Iterated PD also allows researchers to study whether a given incentive structure is stable, i.e. doesn't change much over thousands of iterations. According to the article, the incentives that give rise to police corruption are stable, but tweaking the constraints in a certain way can disrupt them.
The simplest version of the Prisoner's Dilemma succinctly explains both (1) why rational beings feel compelled to have a moral code in the first place, and (2) why they so often disobey the moral code that they themselves believe in. It's best for everyone to cooperate; but in the absence of a guarantee that everyone else will, it's in my best interest to defect.
Of course, you can't get from PD to police corruption in a single step. Lots of intermediate steps and computer modeling is needed to get there. But most of that is probably too technical for an Aeon article, so the article only mentions the first and last steps. A more detailed account of the models they used would be very interesting, but that's probably the job of actual scientific papers.