So you're really in favour of having a Warlord in every part of the city with the different warlords fighting wars between one another? Some rich guy will buy himself enough mercenaries to try and control as large an area as possible, the mafia can just operate more openly and the citizens will have to dodge bullets and shoot back to prevent it. Of course you can solve this by fighting back with a militia but you can also die in said militia if the others are too strong, the police force are many, they have a monopole on power, they can draw forces from other cities if necessary etc. which is why it makes little sense to fight them openly, a local militia is a lot weaker than that.
That the police are not patrolling certain areas is a problem that isn't inherent to the concept of a police force, it's about how our police force is used, handled, trained, equipped and financed. In countries were there is no effective police force that establishes control of an area to keep the peace, it's usually the poor who are at the whim of marauding bandits, warlords and what not, now you can say that all our middle class citizens can afford a gun, but then when I murder someone two blocks away at night, who is going to find out it was me? There is noone to stop me from murdering again, noone who knows how to find me, there will just be a lynch mob that chooses the most unwanted person in the area, perhaps a stranger, and blames, then kills him.
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