In the Return of The King, we see that Elrond doesn't seem so enthusiastic to re-forge the sword. Why is that?
In this scene, Elrond shows why:
In text, the relevant conversation is as follows:
This conversation shows exactly why he refuses to reforge Narsil, the blade of the Numenorean king of Gondor:
That is why he rejects Arwen's pleas at the start:
To Elrond, Arwen's request is nothing more than a desperate plea for her lover, a futile endeavour considering Aragorn's perceived unworthiness to wield the blade of the king, and the weakness of his blood before the might of Sauron.
It is only when he realised that Arwen has chosen the mortal life of Man, that there is no more ship that can bear her west to Valinor, that his daughter's fate is now tied to the fate of Middle Earth, that he finally decided to reforge the sword (and set out to persuade Aragorn to put aside the Ranger).
Elrond didn't reforge the sword because his faith in humanity was restored. He reforged it because doing something, however futile, is better than doing nothing when it comes to saving his own child's life.
In the movie, reforging the sword would draw his (Elrond's) daughter to her one true love, and thereby losing her eternal nature.