I am a Police Constable in Scotland, UK, and the general opinion of where the word “cop” came from is Constables On Patrol.
The UK has been credited with revolutionalising modern policing, as the staff writer says about bobbies on the beat. This literally meant police officers on every street corner, hence, Constables On Patrol.
But it’s not old enough. No one can find any evidence for it until long after “cop” had become common language on both sides of the Atlantic.
And it’s a rule of thumb in English that all explanations of words as deriving from the initials of a phrase are wrong, unless they are no older than the Second World War.