My initial point being that they did state sanctioned violence, which was their role, ceremonial and otherwise.
Only insofar as state sanctioned violence is here defined as "any violence acknowledged as legitimate by the state", which would make every use of self-defense in a modern context also state-sanctioned violence.
So: the association with being a direct embodiment of state authority and power of violence.
So let me get this straight, just so I know what I'm arguing against - your definition of police, in this context, is anyone symbolically associated with the use of violence legitimized by the standards of the polity, even without any attendant legal power.