TL;DR: The "Anger Core" is also referred to as the "Emotion Core", and in psychology, emotion does play an important role in decision making; this may have been driven into a fit of rage and insanity after the prequel of the Portal series, ending up as the "Anger Core". Or, it was simply a gameplay mechanic.
The longer answer:
From the wiki (emphasis mine):
The last core attached to GLaDOS is the anger core. It has a red "eye" with eight dots beneath it. It is also referred to as the "emotion core" and the "aggressive core". Rather than speak, the Core snarls and growls ferociously at Chell. (These snarls and growls were later used in Valve's game Left 4 Dead, which featured the same actor, Mike Patton, as the infected.) Once Chell puts the anger core into the incinerator, GLaDOS begins to implode. In Portal 2, GLaDOS, due to the loss of this core, sounds very mellow, and barely expresses any emotional indications of hostility; she remains resentful over her attempted destruction and makes snarky comments towards Chell while being tested early in the game.
Also the wiki has some additional trivia about the Anger Core (emphasis mine):
When taken near an incinerator, the Anger core will whimper like a dog, signalling fear.
It might be a bit of a stretch to say that these two dots are connected, but in psychology, emotion plays a very important part in making decisions. I recently learned this after reading a book about hope.
To summarize: After having a tumor removed from his brain, a man that used to be very friendly, smart, and helpful, became a witless, non-sensical fool. Turns out that part of the procedure to remove the tumor damaged the part of his brain that created/interpreted emotional responses to things. When interviewed about his decision making process, he was able to follow and explain the exact process he went through when making terrible decisions: the problem was that he simply didn't care - he had no response to how the decision and outcome affected him.
In the same way, it could be argued that the same thing applies to GlaDOS; she was an "emotionless" computer, thinking only about numbers and calculations, but the Aperture Scientists needed more from her than that, so they gave her an emotional response to the results of the experiments of the Portal Test Chambers. From there, it developed into something more - a hatred for humanity maybe? Particularly after implanting an "itch/scratch" response to successfully getting a testee through a chamber (as outlined in Portal 2), that would likely help to build up some rage.
Or, it could simply be that destroying the "Anger" core is the final thing that is driving GlaDOS down the path of death and destruction. The morality core was the only thing keeping her in check, but that was only putting a Band-Aid on the problem: the problem was that the Intelligence core had gone crazy, and the "Emotion" core had buried itself in hatred - therefore the only real solution was a hard reset.
