There is a world of difference between an 'old British bobby' and and American police officer.
The original Parish Constables were citizens employed to ensure that persons summoned to court duly appeared. To that end, they were gven a warrant authorising them to arrest the person and compel them to attend. The warrant also authorised them to raise 'a sufficent body of men' to pursue a fleeing defendant or to overwhelm anyone assisting the defendant in resisting the summons. Be it noted, these were civilians at a time when the local Sheriff, Squire or Baron had a force of men-at-arms at his disposal which couldhave been used forthis purpose, but were not. The Lords' responsibility extended to defence against foreign invasion and dealing with the gangs of outlaws and masterless men that roamed England at that time. In later centuries, the Constables also became night watchmen and acqured the duty to 'keeping the Kings' peace'. To do this, they relied on their fists and staves or batons -no other weapons - and the rule was still one Constable per parish or borough.
It was on this basic structure that Henry Fielding developed the Bow Street Runners in 1749, and which Robert Peel built the Metropoitan Poloce Service in 1829. Peel was very insistent that the new force be a civilian one, hence the decidedly non-military cut of the traditional uniform, and the use or ranks such as Inspector or Superintendent, rather than military ones, as well as the principle of unarmed policing.
This is opposed to the US system, which has had a military flavour from the beginning with military-styke ranks and chains of command. The unfortunate, misunderstood or poorly-worded, Second Amendment added to the problems by producing an armed citizenry with a militia mindset. This forced the police to be armed, created the assumption that all potential suspects would be armed, and thus a tendency to employ deadly force as a first resort.
So yes, the failure of American culture to achieve a true separation between the military ad civilian has caused as much trouble as its failure to enforce separation of church and state.