Okay, so if you are going to have the gov't/public establish the entirety of the hiring/training rules, then what is the point of having a private entity run it? Almost every city currently has underfunded social work, homeless outreach, anti-drug(rehab), and mental health departments in some form. Defunding the police means sending more money their way and adding them to the 911 dispatch hotline. It also means shifting funds to proactive crime reduction endeavors like education, job placement and training, and youth centers. Nothing need be "radically different." Police need to be members of their community like they used to be. They need to actively engage with that community as people even when no crimes are occurring. People like to make fun of "Mayberry" policing as something that can't exist, but it has existed before. The only "different" thing it needs is to be more inclusive as to what defines a "community" and who belongs in it. Demographically, police should also match the neighborhoods they patrol.
Policing is inherently a gov't/public action. It needs to be fixed, but it needs to remain in the hands of the gov't. The unions can and should be broken. Perhaps we should also have elected or mayor appointed mediators/judges in charge of disciplinary proceedings and have jury duty for these proceedings as well. We can even apply this across the board to public workers. Everyone likes to bitch about teachers unions too, and bad teachers getting paid while sidelined. The same thing can happen there too.