We should add that for most of his life the *last* thing Aragorn has wanted has been to be recognized as the Heir of Isildur - especially by the Enemy. So all else aside it wouldn't make a lot of sense for him to make a regular practice of carrying an extremely distinctive broken sword around with him.
I stick to my original hyposthesis, Aragorn has Narsil with him for some special reason not directly connected with the Hobbits or the Ring. The obvious suggestion is he used it to identify himself to somebody or something who presumably owes one to the Isildur's Heir. *Not* I think to Frodo since Gandalf's letter makes no mention of a broken sword - just Aragorn's real name - and so Narsil wouldn't mean anything to the Ringbearer.
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