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...but I am Aragorn, and those verses go with the name." He drew out his sword, and they saw that the blade was indeed broken a foot below the hilt. "Not much use is it, Sam?" said Strider. "But the time is near when it shall be forged anew."
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So even Aragorn admits that Narsil is not much use in the shape that it's in. And despite its origins and history, a foot-long stump of a blade isn't going to do anyone much good.
I can only speculate that Aragorn was not carrying around this sword for most of his time of journeying. It probably remained safe at Rivendell. I suppose that Gandalf asked him to carry it as a sort of "calling card" to show Frodo. After all, a warrior carrying nothing but a broken sword would be a pretty good marker. Still, you would think he might have been carrying some other form of weapon, though Tolkien doesn't refer to it.
I noticed that P.J. avoided this whole conundrum by elimnating any reference to the sword until they got to Rivendell.