In what ways are the existing police services "capitalist"?
All of the police services that I'm familiar with in places like the US, Canada, and Europe lack the fundamental traits of capitalism.
For example, they're the creations of centralized, monopolistic (and often quite socialist) governments, and funded with taxpayer money, rather than arising from the private sector.
As such, there's no competition. Residents and visitors don't get to individually choose which, if any, police service(s) they'll voluntarily receive services from, and what compensation they'll provide, if any. They can't just switch to some other police service if they're dissatisfied with one they've already been choosing to use. They can't unilaterally opt out of dealing with a particular police service.
Entry to the market is also extremely limited. An individual can't just declare himself to be a police officer, and then start offering policing services. An organization of such people can't declare themselves to be a police force.
Unionization of police officers is quite common, as well.
While private security may be available in some areas, it's typically very limited, with no real ability to intervene. They merely monitor an area for trouble, and alert the government's police service(s) if anything arises.
The problems with policing today seem to me to be due to its highly-socialist nature.