That’s why death reportings tend to use “died with X”. I understand that there is a sensitivity to ascribe liability or fault. But if someone dies in recovery from a surgery, or worse under anesthesia, that seems like something that could be identified and reported.

We do (for example: https://www.cdc.gov/healthcare-associated-infections/index.h...), but there's significant difference between "malpractice" and "died due to surgical/anesthesia complications". Perfectly administered anesthesia is still risky, which is why we don't give it for, say, mole removal.