I don't think that the scene of Elrond giving Aragorn Anduril is a flahback in Rivendell but is in fact in Rohan. To me it looks like it could be in a tent on the battlefield somewhere. In FOTR Elrond made the offer to Aragorn that Narsil be reforged but Aragorn refused saying that he did not have or want the power to wield Narsil. I don't think that Elrond would've had it reforged anyway and offer it to him again after he refused it once before.
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His sword was long his lance was keen
His shining helm afar was seen
The countless stars of heavens field
Were mirrored in his silver shield
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