Apart from violence, there's another big thing that probably went missing, too - authorization for certain people to enter private property for reasons of the general good. The reason we call the police for welfare checks, for better or worse, is that nobody else has the right to enter your house. A doctor might be better suited to responding to someone undergoing a mental health crisis, but they can't break in. Similarly, you see stories of "police rescue deer from rooftop" or whatever because nobody else is authorized to climb onto random rooftops. If a society wants to get rid of the police, it needs to designate some other group to handle this use case. It can't simply get rid of the police.
A "natural experiment" of a world without police is quite unnatural: it's a world built up around the police with a sudden police-shaped gap in the middle.
To pick an analogy that should make sense to folks here, it's like shutting down your datacenter for 16 hours, suffering serious outages, and then concluding that your company absolutely needs its datacenter. Well, yes, it does today, but that's not what the people saying you should look at public cloud are advocating.