We should have police forces with monopolies on violence (with specific exceptions) because inter-personal violence is a bad way to resolve disagreements

One issue is, and I'm not even saying you're wrong, but the police aren't there at the scene...but people are. Police were never intended to prevent crime or protect people from direct action; they're meant to arrest criminals. They were not meant to replace people's right to self defense. Which is not to say that this was all self defense, but you get my point.

Police do have a purpose, but to say they should have a monopoly on violence (even with exceptions) is not quite as cut and dried as it might seem.