I would argue that there is a difference between what the primary responsibility of police officers should be and what it actually is.
I believe that the primary responsibility of police officers should be to prevent crime. We, as members of the public, do not want to become the victims of crimes. It is much more important to us that crimes be prevented and somewhat less important to us that crimes be solved once they are committed. Therefore, we would prefer that the police be able to prevent us from becoming victims of crime in the first place.
While this should be the primary responsibility of police, their real primary responsibility is often determining who committed a crime that actually did happen. It is often very difficult for police to prevent crimes given the resources that they have. Therefore, they end up in more reactive positions, being asked to determine who committed crimes and to apprehend those people. While this is a very important task, it would be better if the police could prevent those crimes from occurring at all.