But that's a perfect example of something we don't actually know.

Yes, physicians improve their metrics when their individualized data is public. But is that because they are improving their care, or because they are adjusting their practice to game the system. (e.g. surgeons choosing to not operate on sicker patient so their mortality numbers improve.)

For example, in my hospital, we have a location in every hospital room where the estimated date of discharge is supposed to be written and updated by the docs, to help patients plan. When the hospital started auditing and making "public" (known to the rest of the department) how good individual docs were at keeping this date updated, you know what happened? Every doc started writing "TBD" as the estimated discharge date (even those who previously were more specific), because the auditing process considered that acceptable, and everyone wanted their numbers to look good. Or at least, not look the worst.

So what started as an imperfectly adopted but very reasonable intervention to incrementally improve patient care turned into a total waste of time for everyone.