Somewhere in the legendarium, it's said that the Eldar believe participating in war, fighting, violence, etc. diminishes one's innate healing ability, and therefore those who want to focus on healing may prefer to avoid fighting when possible.
But I would draw your attention to an interesting tidbit from The Return of the King. Aragorn comes into Minas Tirith, without yet revealing his full identity, to heal the many injured after the battle. It's specifically noted that he was accompanied by the sons of Elrond, and together they worked long into the night.
That strikes me as interesting. Almost one could extrapolate from the little lore on Elladan and Elrohir that these twins were kinda violent people, craving vengeance on the orcs. Yet there must have been something that preserved their healing powers. It's not directly stated that they inherited their father's abilities in that area, but it seems implied.
So if the sons of Elrond can be fierce warriors and yet also healers, it leads one to wonder how it all adds up. Did they retain a measure of healing abilities but compromise them by their participation in violence? Was the belief of the Eldar just a theory or a bit of folklore? And if it was an incorrect belief, what other things did they believe that weren't completely accurate and correct?
I don't think there's a solid answer to this, it's just something I've been thinking about. At the very least, perhaps Elladan and Elrohir could put basic, practical medical knowledge into practice even if they didn't have an innate "Elven power." But maybe there's some key interplay with compassion and its relationship with violence born from defending others, as opposed to violence born of malice and thus void of compassion. Again, I don't know. But I do have to say, I like it that Elrond's sons wanted to take part in healing. I think it shows they had something of their father's caring nature, and that they hadn't been wholly lost to an all consuming madness like other parts of their story may suggest.