No, “mob rule” or community defence is exactly what we need. You’ve been taught it’s a dirty word, it’s not.
The choices of how crimes are handled are best handled at the local level. The people living in the same places as the victims and perpetrators know best how to deal with each instance. Once you absolve that power to a police force people stop taking personal responsibility for their wellbeing and that of their community. The loss of sense of community is the worst thing that can happen for stopping crime.
This already plays out today in places such as the Zapatista autonomous territory in the Chiapas. There they work on a reformation model, with the community patrolling and defending itself, the guilty are still judged and sentenced by their community in fair trials. It’s a far more just system then the “justice” model we employ.
As for violent axe murderers who are irredeemable, those account for 0.3-3% (recidivism rate) of 0.43% (homicide rate) of all crimes.