The examination of crime & punishment in society hundreds of years ago in the pre-police eras of history, and extrapolating that into the modern era, in totally untenable for two main reasons I think: the size and complexity of civilization has greatly increased since ye olden times, and the physical mobility of individuals has vastly increased.
The population of each individual town and city today is vastly greater than it ever was in a pre-police era. This reduces the sense of community, as each person in their life interacts with orders of magnitude more strangers than any person from an olden community. I'm not saying that even a 1000 years ago people knew everyone in their town, the age of tribes where everyone knew everyone are ancient history, but the degrees of separation between people were much closer. For example: 1000 years ago, if you met a stranger, you might not know him, but you probably know someone who's related to that stranger. Nowadays, if you meet a stranger, you might know someone who knows someone who's brother has a friend that knows that stranger. That's basically complete anonymity if the person is determined to not let you know who they are.
My point is this: in the absence of a dedicated police force, the community has to police itself. If the community is so large that tracking criminals and executing justice becomes too onerous for even the angriest of angry mobs, then a dedicated police force is required. I believe this is what happened historically: as innovations in technology made living in large, highly populated cities possible and desirable, the resulting surge in population density required the invention of specialized policing force. (And the further evolution of industrial society into a global modern society probably required the invention of a more sophisticated police system that is able to self-police and expurgate bad actors immediately, as well as uphold it's standards of conduct and service to the highest degree that a modern society requires; but that didn't happen... though now I'm speculating.)
The second reason being the increase in personal mobility. What I'm getting at is that dedicated policing doesn't just create security IN Towns/Cities, but also in every place inbetween; that is, on the roads between towns/cities. In the absence of a police force, there might come into existence a new breed of bandit/pirate that move from place to place, preying on people that are far from any angry mob that could possibly help them, and then quickly move to a new location before any effort to exact justice can be levied against them. In olden times, people were generally much poorer, and had worse forms of transportation, and that's not true today.
There's also the concern that wanted criminals, who might imminently be on the receiving end of a lynching or public ostracization, can just get in their cars and leave town. In olden times, you'd have to leave town by foot or horseback, and you can only get so far, and the total number of places you could possible go are limited. By car however, you could travel to the other end of the United States in a few days if you really wanted to. Unless the offended parties are willing to get in their cars and track the criminal down personally, that person is almost guaranteed to get away with any crime they might have committed.
Now, do I think that "Abolishing Police" (I think that's a terrible misnomer and awful branding; they aren't looking to abolish police, they want to replace them with better and more scrupulous police; but I suppose "Replace Police" isn't as impactful a slogan) is something worth pursuing? Absolutely, the police are meant to be one of the arms of the state that serve the public, and, metaphorically speaking, if my real biological arm suddenly gained a mind of its own and started moving and doing things against my will: like sabotaging me, destroying my things, hurting people or myself, it'd be foolish to not amputate it immediately. Of course, it'd hurt, and I wouldn't want to do it, but in order to even have a chance of living normally and being happy, I'd have to do it, no matter how unpleasant it might be. Then, afterwards, I'd have to undergo some hardship as I learned to cope without an arm, but then I could actually work and go about about getting a replacement arm to fulfill the original function of the amputated arm, without the interference of the malicious arm hindering me.