There are some occasions where we can see healers at work in the books, most of them mentioned by you already... but here we go with some of my views (adapted from a post i did on board which is now far gone and forgotten, luckely i saved the drafts):
The only time we see an Elf heal someone is Glorfindel, who searched the wound [Frodo's] with his fingers, after which Frodo felt a little warmer and his vision cleared. But Glorfindel said "this is beyond my skill to heal." Elrond did much more, but we don't see what or how he did it, only that he "drew forth a splinter." Hard to say whether he used conventional surgery or magic or both, but evidently the wound had healed over by the next day when Frodo woke up. It's interesting to note that both Aragorn and Glorfindel say they don't have the skill to heal it. Probably Aragorn had noticed the broken tip of the knife before it melted, and guessed he could do little without surgery. So for sure Elrond is a very powerfull healer!
The some more about "The hands of the King"
'Here I must put forth all such power and skill as is given to me,' he said. 'Would that Elrond were here, for he is the eldest of all our race and has the greater power.'
It's hard to tell which parts of Aragorn's healing are
1) Númenorean medicine
2) Elvish medicine, no doubt taught to him by Elrond
3) something innate and peculiar to the King, which is a common motif in certain "king archetypes".
However, that quote suggests Aragorn has enough skill/training to match Elrond; his power is just less. Power from where? If Elrond is the "eldest" of Aragorn's race, then it's not Elf or Mortal to whom he's referring, but to descendents of Lúthien who inherited Maiar blood from her mother. So also here we are on the same track! (like minds think alike hehe)
The we see the healing of Frodo on Weathertop: Aragorn sings strange words over the Morgul-knife, whispers something to Frodo, and then bathes the wound with athelas. Hard to say whether this is Númenorean (is he singing in Adunaic?) since he's using an herb of Númenor, or whether this is Elrond's training.
Frodo, Sam, Gimli are all healed by Aragorn during their journeys. We don't see how Aragorn healed Gimli at Helm's Deep, but for Frodo and Sam's bruises and gashes after the Moria-fight, Aragorn simply gives them first aid and again bathes the wounds with athelas. No singing here: that was evidently a response to the evil enchantment on the Morgul-knife.
Healing "The Black Breath": The rhyme implies that the king alone is able to combat it. The Númenoreans who made up the rhyme would have no idea whether Elves like Elrond could also combat it, but Aragorn's comment about Elrond implies that yes, he also could deal with this.
Later when Aragorn heals Faramir we encounter some strange behavior. Before Aragorn gets his hands on athelas, he does go into a trance:
Now Aragorn knelt beside Faramir, and held a hand upon his brow. And those that watched felt that some great struggle was going on. For Aragorn's face grew grey with weariness; and ever and anon he called the name of Faramir, but each time more faintly to their hearing, as if Aragorn himself was removed from them, and walked afar in some dark vale, calling for one that was lost.
Whatever he's doing, he's awake and lucid the instant Bergil rushes in with some leaves. Aragorn seems to have reached Faramir's mind somehow by doing this, because Faramir knows who he is immediately. That could also be because Faramir's got special blood himself. Frodo says he has "an air of wizards", and what are wizards? Maiar!
The healing trance is used to pull someone back from the brink of death. I don't know whether this would work with just anyone, or whether it's because both of them have a tiny amount of Maiar blood. Again, I say Maiar, not Elven, because otherwise Aragorn wouldn't have said it was something peculiar to Elrond's bloodline.
Most interesting maybe is how Aragorn actually does heal people, we see some variations. Let us take a look at them:
Aragorn does something specific to fight the Black Breath: he lays the leaves in his hands and breathes on them. The hands of the king, and most importantly his own breath, are needed to combat this unique malady.
He doesn't even bother to bathe Faramir's wounds; he just holds the steaming bowl up to his face. The scent, the air, is what's needed to combat Black Breath. Faramir instantly wakes up.
For Éowyn he again does something specific for the person to "summon" her, kissing her on the brow before using athelas (also yakking rather a lot before doing anything, silly epic heroes). In Éowyn's case he bathes her arm and forehead. He can't reach her spiritually as he did Faramir; her bond with her brother is stronger, so Éomer much call her as Aragorn did the Steward.
We don't see what he does for Merry (or Frodo and Sam at the Fields of Cormallen), but again, it appears that the treatment varies widely depending on the spiritual/mental state of the patient, as well as the nature of the wound. If the patient's spirit is resilient, which is a special trait of hobbits, they'll bounce back easily, but if the patient has taken a bad blow, the healing won't work (Éowyn, Celebrían). And it requires both "power" and "skill". The skill is that Aragorn knows how to vary the treatment according to circumstances.
Ok, so far my views, anyone else has more on this?