The police in this case knew the name of the victim, the name of the perpetrator, and had the ability to look over the evidence to figure out whether a crime had been committed (according to the statute, it probably had). You don't get much more open and shut than that. Yes, it takes resources with only modest results (modest in the sense that it's not usually gonna be a murder case or a drug smuggling case, and won't be a gold star on the officer's record, but they will probably be able to help someone), but so does so-called "broken windows" policing and stop-and-frisk, which have similarly modest results (and have the unpleasant side effect of creating new victims in many cases, and breeding mistrust for police among young brown folks in nearly all cases).

I'm not sure what's so hard about this. Police in the first instance should at the least take a report from the victim. That's the first step toward a restraining order and other legal remedies, and how things begin in other kinds of harassment cases.