Nursing—the art of keeping patients clean, comfortable, and cheerful—must have saved far more lives than doctoring in the long dark ages before antibiotics. Florence Nightingale (a significant mathematician, by the way) has to be reckoned one of the great benefactors of humanity.
Surgery before modern anesthesia (which arrived in the 1840s) was a horror show. Fanny Burney’s account of her mastectomy in 1811, when she was 59, is hard to read. She survived it, though, living to be 87, and even stayed friends with the surgeon.
The main component of the surgeon’s art was speed—getting the job done before the patient died of shock.