Let me suggest such a metric: A T-value. That is, the degree to which the actual outcome deviates from the expected outcome.
For example, say you have a hospital that does stem cell transplants. For each patient before the treatment you assess the "chance that the patient will die within 1 year" based on that patient's age, sex, BMI, heart and lung function, type of disease, time since last relapse, quality of donor match, etc. From this you estimate that 19.6% of the hospital's patients who are treated over a particular period will die within 1 year, with a standard deviation of 3.7%. The actual mortality rate for this group of patients turns out to be 26.2%. So the T-value is +1.78; this is the 'single metric' used to evaluate the competence of the hospital.
> there's the problem of deference to eminence combined with small in-bred communities within each speciality or sub-specialty (who would likely be the only people with the training to evaluate their peers reliably).
Keep in mind that expertise is only required in estimating pre-treatment "chance that the patient will die within 1 year". Determining whether a patient is alive after a year and calculating the T-value can be done by anyone. And even when estimating expected mortality, there is plenty of evidence that a simple algorithm can actually beat the experts - Daniel Kahneman devotes a whole chapter to this point in 'Thinking Fast and Slow' [1].
[1] Chapter 21: https://www.amazon.com/dp/B00555X8OA/ref=dp-kindle-redirect?...