I'm pretty sure this is "A Touch Beyond" by Stephen L. Burns, published in Analog, January 1985. There is a sequel, "Angel" featuring the same protagonist published in Analog, January 1990.
The protagonist, Dr. Marchey, is a "Bergmann surgeon."
This lead to an organization that would take apprentices and put them through some sort of grueling training/process where at the end of it they deliberately had their hands amputated and if the process worked they'd become doctors/healers.
"Listen Ella, there is a phenomenon common among amputees called the 'phantom limb.' That means that they can still 'feel' the missing limb, that something remains although the limb is gone. The strength of the feeling varies from person to person; some do not experience it at all.
"A very great man, Dr. Saul Bergmann, studied this phenomenon. He found that less than a hundredth of one percent of those who felt that phantom limb could actually manipulate matter with that limb image. The ability was so weak and wildly erratic that it took him several years to prove conclusively that it existed. But he did prove it, and also proved that with practice and special training this ability would grow stronger and under better control."
"There was no accident. He took the Bergmann Tests and scored high enough to be considered. He took the preparatory training, and once it became clear that he had something of the innate ability needed, he gambled on success and had his hands amputated. God, I'd trade mine if—"
These healers would be given cybernetic prosthetics for hands, basically bionic hands to replace their amputated ones, which functioned normally.
He began to strip off one gray velvet glove. The fabric came away and underneath it his forearm was silver. His wrist was silver. His hand—palm, thumb and fingers, was silver; metal sculpted and polished, metal shaped and jointed to mimic the flesh and bone it had replaced. He pushed up his other sleeve and removed his other glove, his already-revealed hand winking in the light as it moved like a thing alive. His other hand and arm were the same, a mirror-twin of the first.
During a procedure their prosthetics would either retract or possibly they were removed entirely.
He bent like a badly made puppet and rested his forearms on the table; palms up, elbow to wrist flat on the padded surface. The clockwork birds of his hands did not move. His awful, unreadable eyes closed. He drew his breath through his clenched teeth sharply, as if lifting an impossible burden.
His breath came out in a long hiss and he stepped back, straightening up slowly.
His silver arms still lay on the table, lifeless and abandoned, gleaming and
somehow obscene. Just below the joints of his elbows his arms ended in featureless silver plates. Below that there was nothing.
Dr. Marchey has been called to treat a young girl who critically injured herself playing with homemade fireworks; she has metal fragments in her brain, a badly damaged eye and extensive damage to her face.
The healer would then reach their invisible ghost hands inside the patient and begin the healing process.
Dr. Chang turned to look, just in time to watch a jagged fragment of metal slowly emerge through the gauze covering one of Shei's wounds. It poked out apparently by itself, twisted free of the threads, hung there a second, then lay on the white bandage. A small bloodstain spread slowly away, darkening the clean gauze.
I'm not sure if they were performing surgery, but I think they could do all sorts of things like accelerate cell division, repair tissues, cure diseases, etc.
She had seen the bandages come away from Shei's face by themselves, revealing the lacerated, hastily sutured flesh beneath. She had seen chipped bone replace itself and damaged veins snake together and reknit. She had seen a damaged eye become whole again, had seen subcutaneous tissues move like hot wax, flattening, filling, and sealing. She had seen lacerated—shredded, and burned skin reshape and return to its earlier state as water stills after a disturbing hand is withdrawn. At the end of it Shei's face showed nothing of the gross insult it had suffered.
There was someone with the healer that might have been a trainee or other observer, but that's all I can recall.
The story is told mostly from the points of view of Dr. Marchey's former lover Ella Prime and the attending doctor Dr. Chang, both of whom are present for the first part of the surgery.
Note it's possible you're recalling "Angel" instead, though it's outside your time period. Dr. Marchey's patient in "Angel" is a dying cult leader who is past saving, which matches your question, but there's no observer in this case, and less description of the what and how of a Bergmann Surgeon. Taken together with your timeframe, I think "A Touch Beyond" is more likely.