If the plan really is to somehow kill off the entire police force, it’s really only going to create a power vacuum that could very likely be filled with a designated national military with shoot-on-sight orders, if some other type of murderous psychopath doesn’t slot in their militia first. Those who are looking to murder any and every officer they come across probably don’t have a robust plan either and will only contribute chaos, regardless of how “successful” they were.
I could only see some type of vigilante force pushing the designated police forces out of power to be effective, but it wouldn’t be pretty and it would also have a considerable chance for corruption. Vigilantes both act for the law and outside of it, and would have to self-govern. If one vigilante is taking things too far, it would be up to the others to end it. On the other hand, if all the vigilantes act out of where they were supposed to, the people would have to dispose of them, and it returns to effectively cop killing once more.
If there is some kind of bound honor code that the vigilantes follow, it could strengthen bonds between them, but it also needs to not be bent. If they agree to only carry out executions for the most heinous of criminals, they can’t show bias because it would become obvious. To “streamline” the justice system into a single force is a very risky move, but would keep cases from being held in waiting for so long, and arguably assist with prison crowding, though prisons themselves may need to be redesigned to fit such an adaptation, since not every criminal would be getting executed on the street if the system was even remotely balanced.
The obvious comparison is the Judge system from 2000 AD, if it would come to a point where a brutal enforcement system from a gritty comic series ends up being necessary in real-world society. The idea is to reduce brutality, even if the primary step in the way is to just arrest everyone who doesn’t have a proper hamster license or whatever minor infractions they incur in daily life. A vigilante force should really only be looking out for the worst of criminals, the most dangerous to everyone, even if they’re within another law enforcement sector. Perhaps the vigilantes would mostly be there to police the existing police while they still exist, and to ignore the double jeopardy clause if they feel that the justice system made a mistake, but perhaps to not just enact revenge if the criminal truly becomes no longer a threat, just not stuck in prison. Most importantly, a vigilante should not reignite a defused situation, such as if a criminal took hostages but didn’t kill any and surrendered without any force needed.
Ideally, peaceful resolutions would be easier to come by, but the primal human instinct is violence, and is why war never ends regardless of its scale. Those who cannot be violent themselves just instigate others to do it. Capital punishment and execution seem likely to make a comeback, but the question is for what crimes, and who would carry it out.