Narsil wasn't the only famous sword to have been broken, but the one most likely to be known of in Gondor. They probably didn't have a functioning memory of Ringil, as an example.
It seems unlikely that news of Isildur's fall, with the delivery of the shards of Narsil (heirloom to the house of Elendil) to Valandil wouldn't have made it to Gondor. If presumed lost, it would surely be in the fall of Fornost. Perhaps it is more thought of as lost to Gondor, as Arnor held them and Arnor is no more.
The focus on the moment of reunion of the shards of Narsil and the Enemy's Ring only makes sense in retrospect. At the time of receiving the dream poem it is opaque.
The prose gloss of the poem with the knowledge available in Gondor seems to be:
Go look for a sword that was broken, it's in Imladris, get some really good advice to help with your current problems, you'll know this is happening when a token is shown that 'doom' is close, as the thing that killed Isildur will 'waken', and the 'halfing' will 'stand forth'.
Even this is still opaque, because 'doom' (whose and in what sense: fate? decision?), 'waken' (in what sense, and what does that mean?), 'halfling' (in what sense: mixed blood? small stature? half-witted?), and 'stand forth' (bear witness? distinguish itself? stand in front of?) are all ambiguous.