Specifically, I can't understand why a coroner would not take actions to facilitate the prosecution of a crime (infanticide is murder), because that is one of the jobs of a coroner.

Because he, like some who have posted here, is sympathetic to the baby-killing mothers under certain circumstances and doesn't mind helping them avoid prosecution? I wouldn't judge him, heavens forbid. I'd likely do the opposite in his place, but I respect his position.