An “outstretched arm” is the sword, as it says “and his sword was unsheathed in hand, outstretched over yerushalayim” (Divrei Hayamim 1 21:16)
Where was a sword present in the plagues of Egypt? A variety of answers have been proposed. Perhaps it is a reference to the threat at the beginning (Shemot 5:3) which was never fulfilled, but was a punishment through causing great fear. Or perhaps a reference to a civil war between the firstborns and the other Egyptians when they heard that Pharoah would leave them to their fate, and their parents wouldn’t do anything about it. But whatever the case, how does this emerge from the Pasuk in Divrei hayamim?
The Pasuk is a description of the pause in the plague which was threatening the Jewish people in the aftermath of David counting the people. What is the metaphor of an angel with a sword that was outstretched over Jerusalem? The Rambam (Moreh Nevuchim 3:22) identifies it with the angel of death, which is the same thing as the yetzer hara and the satan. This is also expressed in the story of Bilaam and his donkey. It points to the fact that matter and fantasy are the sources of harm. The Satan, which directs us off course is a feature of our physical embodiment. Thus our fantasies and unrealistic desires are the sources of most of our troubles. Ultimately “no evil comes from above” and we impose most suffering on ourselves.
In the time of David, they realized that if they continued in the error which lead to the plague, Jerusalem would be destroyed. The only way to save Jerusalem was to use the korbanot to redirect our psychological needs and elevate them to being used in serving Hashem.
This is the root of the various explanations. A civil war is a self imposed harm. The plague itself would only have done a certain amount of harm, but the fantasies governing each group led to members of society harming each other. Similarly the idea of the unfulfilled threat. Their imagination got the better of them and caused them to panic. The only harm it did was due to their own apprehension, as opposed to a more realistic evaluating their immediate situation and not panicking about an unknown future which they have no control over.
This “sword” of complete destruction was recognized by the Egyptians during Makat Bechorot, which, like in the time of David, they realized was only a pause in the destruction, and that the whole Egypt would be destroyed if they did not correct their ways. If they did not do Teshuva at that moment, they would all be destroyed. They finally released the Jewish people when they realized their very survival was at risk.