Actually let's do better. Disband police and require that every adult citizen without severe mental difficulties or violent criminal record be trained in firearm use and be required to carry a weapon. If there is a wrongful shooting then there should be legal repercussions. Being ordinary people that's more likely than for police, right? The conditions for proper use of lethal force would be a "reasonable standard" test. In other words if a panel of citizens finds that an ordinary person could have reasonably believed that lethal force was required to stop a criminal act of sufficient seriousness, or likewise, that their life or that of another required the use of deadly force then no charges follow.
Since you aren't a pussy I expect you would endorse such a plan. Crime would plummet, police don't die or get away with killing, ordinary citizens wouldn't have a police union to protect them.
A perfect solution for your requirements.
I could name your solution "The Wild Wild West Scenario".
Actually, your solution has some unintentional circular logic to take into consideration:
Under your hypothetical scenario, it's likely that anarchy will follow. The criminal element will naturally take control, due to their willingness to go a step further than any normal citizen would no matter how much higher things escalate and the fact that the criminal element are already well armed and well positioned to take control of the territory they're located in.
Cries for the police to take back the streets from the criminal element will be heard, martial law is declared, the criminal element go back to their old ways, the cops go back to their old ways, some kind of order is restored for awhile and the whole process repeats itself until the real reasons for having this cycle exist in the first place get taken care of. And that will never happen as our politicians are more concerned with keeping a hold on office via appeasing the folks who really matter: the very rich who the pols MUST suck up to, and whoever else is left, other than the folks who don't vote, who just so happen to be the folks that suffer the most from the aforementioned repeating cycle of despair and abandonment.