To the OP, possibly Tolkien didnt put much thought into how he proposed to have Narsil reforged, if you weld a blade back together it will fracture at the welds in combat and you destroy the temper of the blade, if you destroy the temper of the blade your pretty much ruining the "Telchar" in the blade, if its retempered its no longer a blade made by Telchar.
If you suppose the magic in the blade is inherent in the metal, maybe its mithril or galvorn, then if you melt it down then beat it out into a sword shape then the nature of smithing means you will get slag and metal being lost in the forging process thus you get diminishing returns and a longsword sized Narsil would become a short sword sized Anduril unless you added extra ingots to your smelt which would mean inherently your new blade is not the same as the old blade.
If a elf smith in Rivendel can reproduce a blade with the same balance and temper as Telchar from some some recycled ingots that performs better than narsil (anduril was more potent than narsil?) then why isnt he mass producing them?.
Of course Mithril and Galvorn may not have steel equivalents and the introduction of carbon may not be necessary thus there is no tempering required, and maybe theres some elvish super glue in Rivendell that was the ideal solution to the problem.
PS I always thought Telchar sounded more Sindarin than Dwarvish, maybe this was what the Sindar called him.