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I think Martinez overstates his point here.
Certainly Narsil/Anduril was the most *visible* token of his heritage which Aragorn wielded - though perhaps just as important were the (victorious) sinews which wielded it as well.
Most of the heirlooms of the House of Isildur had been lost by T.A. 3018 - but not all. The Sword aside, there was also the Elendilmir, the token of royalty in Arnor, worn by every heir of Isildur since Valandil. Also there was the Ring of Barahir, passed down from that famous hero of the Edain down to Numenor, to the Lords of Andunie and their heirs, the House of Elendil - the only artifact to survive the drowning of Arvedui in the Ice Bay of Forochel. The Rod of Annuminas (the other token of royalty in Arnor) Elrond of course had withheld - and perhaps Martinez is under the impression that these other heirlooms were withheld as well by Elrond. That is not clear to me from the text.
Establishing Aragorn's lineage - even in the absence of a Mormon Family Library network or a (formal) Arnorian House of Arms or somesuch - would have been possible, with enough personnel (Gimli, Legloas, the hobbits, the Dunedain of the North, Elrohir and Elladan) on the scene to establish his bonafides and more available to be called on in Rivendell, Eriador, and Lothlorien. Ultimately, however, Aragorn's claim succeeded where Arvedui's failed because of his feats on the Field of Gondor (just as Earnil's paved his own way to the throne) and the force of his personality, as much because of the lack of a valid claimant from the House of Anarion.