As people have pointed out before, the phrasing of Glorfindel’s words about the Witch-king allow for quite a number of the inhabitants of Middle Earth to kill him, if we’re only looking for possible loopholes in the prophecy.
not by the hand of man shall he fall
According to this, the Witch-king could technically be killed by elves, dwarves, ents, hobbits, orcs or maiar. Why doesn’t Legolas kill the Witch-king? Why doesn’t Gandalf?
As mentioned, elves are very aware of the story logic that governs Middle Earth. They see their own place in the narrative, they know which foes are beyond them. Gandalf, too, knows that he cannot be the one to kill the Witch-king, and the Witch-king knows that Gandalf cannot kill him. Through their combined beliefs, the outcome of their fight is predetermined.
Eowyn doesn’t know what she can or cannot do according to story logic. The Witch-king has killed her Theoden. She sees no reason why she shouldn’t avenge him. And when she hears the Witch-king tell her that no man can kill him, she simply decides that that rule doesn’t apply to her.
Eowyn isn’t the only person who could have been the exception to the rule, but she is the first person who decides to genuinely, honestly believe that she is the exception to the rule, and this is why she ultimately kills the Witch-king.
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