An interesting point -- it seems like everything surrounding Bilbo, Frodo, Aragorn, the Ring, Gandalf is on a strictly need-to-know basis, Gandalf having made the point that treason has always been the "greatest foe."
Something new that struck me on this reading was the nature of the friendship between Bilbo and Aragorn. Bilbo is comfortable enough with Aragorn to tease him about standing up Arwen at the feast. He tells Frodo he gets more Shire news from the Dúnadan than from Gandalf. We also learn later that Bilbo is the one who composed the “All that is gold” poem which even Gandalf uses as an identity check in his letter.
A search of my Kindle version shows that “Dúnadan” is used very rarely in the text:
Glorfindel - once
Halbarad - once, identifying himself in Rohan as Halbarad Dúnadan
Aragorn - twice, with Éomer when they first meet, and with Pippen at Isengard
Bilbo - 6 times, mostly as “the Dúnadan”
Bilbo says that ‘he is often called that here’ in Rivendell, but it is certainly a soubriquet that Bilbo has enthusiastically adopted. Is it a form of teasing? Does Aragorn call him ‘Halfling?’ And exactly how well do they know each other?
According to the Tale of Years, Gandalf first confided his doubts to Aragorn in 3001, soon after the birthday party, but by 3000 the Shire was already being closely guarded by Rangers. Bilbo settles in Rivendell in 3002, and while Aragorn is helping Gandalf off and on for the ensuing years, he certainly used Rivendell as his home base (especially after Arwen returned in 3009). That’s sixteen years with both living in Rivendell, with Aragorn gone for long periods of time. I don’t like to think that Gandalf would ask Aragorn to cultivate a friendship, but I am sure that Aragorn would think it important to know as much as he could about this former Ringbearer, especially as Bilbo did what Isildur could not do.
Moreover — and this is something I so much wanted to see in the Hobbit film — when Bilbo first stayed in Rivendell, he would certainly have noticed a 10-year-old human boy. According to Karen Wynn Fonstad’s calculations, he was there about 3 weeks — which in my head-canon is certainly enough time to strike up a friendship with someone his own size (à la Pippin and Bergil). Picture Bilbo entertaining Estel with the story of the Trolls. What a lovely surprise to discover that boy’s true identity 60 years later, after which I can well imagine Bilbo taking a proprietary interest in all things Estel/Aragorn/Dúnadan.