[The Squatter of Amon Rûdh]

I'd be very surprised if anyone picked up everything in LoTR on the first reading, simply because it's such a huge story. There's a lot to take in there, and more in the other books. It amazes me how much people on this forum know about the whole corpus.
I don't think that I noticed the sub-text between Eowyn and Aragorn either until the current re-reading, although I see it more as an infatuation: Aragorn's a hero out of legend; brave, wise, noble and a potential king, who's fighting the Great Enemy without obvious fear; Eowyn's an adventurous girl, who's been nursing Theoden though his dotage, which isn't a great deal of fun, and feels trapped by her various frustratingly boring duties. Adventure and escape are what she really wants and Aragorn's just a focus for that desire. When she's had her fill of adventure, her interest in him must naturally wane. Anyway, the amusing part is that you can see the exact moment when the penny drops and we actually see Elessar, Isildur's heir embarrassed, suddenly realising that there are more perils facing him than those dreamed up by Sauron.