So this was bumped...and then it wasn't. But in the interim I saw it and did some googling.
My apologies if it was posted earlier in the thread (I only skimmed) and my apologies if I'm out of order even posting in this thread at all, but this article seemed very insightful and relevant to the topic at hand: Sacral kingship: Aragorn as the rightful and sacrificial king in The Lord of the Rings
IN DISCUSSING ARAGORN, Verlyn Flieger notes that:
The concept of the king as healer derives from the early Celtic
principle of sacral kingship, whereby the health and fertility of
the land are dependent on the coming of the rightful king. Where
there is no king, or where the king is infirm, the land also will
be barren. (50)
The fact that Aragorn is a healer, both of his people and of the land, is an important motif of The Lord of the Rings [LotR] and emerges gradually as the narrative progresses. Aragorn tends Frodo after the attack at Weathertop (LotR I:12 192-94), Sam and Frodo after their escape from the mines of Moria (II:6 326-27), Faramir, Éowyn, Merry and others injured in combat, after the battle of the Pelennor Fields (V:8 844-53), and Frodo and Sam after the fall of Sauron (VI:4 931). Similarly, his ascension marks the end of the lands' despoiling, his resolve that Minas Ithil be "utterly destroyed" in order for the cleansing process to begin (VI:5 948), and the imposition of order on his realm (VI:7 971), all symbolized by the blossoming of the White Tree in the Court of the Fountain (VI:5 951). The notion of healing here is broad, and reflects Strathern and Stewart's definition of healing (as opposed to curing), namely that which refers to "the whole person or the whole body seen as an integrated system with both physical and spiritual components" (7).
Tolkien loved and read many old mythologies, so this is probably where the king-as-healer thing came from.
ETA: believe it or not, my search term was "king as healer" and this article about Aragorn was the first hit!