What was the point?

[Birdland]

...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."

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.