It's never explained exactly WHY a six-minute surgical procedure required (well, nine doctors at all, but especially) Dr. Hargreaves, who is a psychiatrist.
The procedure was probably expected to take much longer, she just died very early in the process. The exact surgical procedure isn't specified - "radical resection" just means that something was being removed - but it's possible that it involved a brain tumor which Hargreaves initially deduced existed from Mrs. Phibes' mental state. He attended the surgery as an observer following up on his patient, and Phibes held him culpable because of his role in the diagnosis.
Just why is it that Dr. Phibes is supposed to be a sympathetic character at all? Yes, his wife died and he was disfigured, but let's be honest: there was never any suggestion of malice or incompetence on the part of Phibes' victims, his wife was just so badly ill or injured that a nine-man surgical team couldn't save her. And if that was the case, a 10th doctor wouldn't have helped at all - never mind that Phibes himself is not a surgeon, but a theologian, a musician and a clockwork engineer, and so he had no relevant skills to offer. The death of Mrs. Phibes was a tragedy, yes, but her doctors did no wrong in being unable to save her, and Dr. Phibes caused the accident that disfigured him himself: his motives and goals are nothing short than those of a madman.