Having flown public use in the past, I have found that much of the misunderstanding is caused by urban legends. All the agencies I dealt with had much stricter requirements than the FAA. A great number of the 'public use' aircraft are not even owned or operated by government agencies. They are owned by private companies and operated under contract to various government agencies. And they still fall under Public Use. The agencies that do the contracting have strict requirements for maintenance, pilot qualifications, crew rest, etc.
As for bring police officers off the street into the aviation division, it makes much more sense to do that than try to teach a pilot all the streets, back streets and alleys. Plus they have their police experience to fall back on when on patrol. The police departments I have talked with do all their training in house and fell they maintain better control that way. And how each geographical area sets up their aviation sections depends on what works for them. And that is the way it should be.