A country built around driving leads to profound effects on what police do:

It is not an exaggeration to say that police power in the United States is built around the unique conditions created by car culture, in which virtually everyone is breaking the law all the time—with occasionally severe consequences. In her book Policing the Open Road, the legal scholar Sarah Seo points out that mass car ownership prompted a wholesale reinterpretation of the Fourth Amendment, which protects us against search and seizure. Or it did, until we all started driving everywhere.
Police often abuse this authority to perform “pretextual stops” hoping to find guns or drugs, knowing that trivial traffic violations give them the power to search citizens at will. Officers have at times undertaken this constitutional sleight of hand with explicit federal endorsement, deputized as foot soldiers in the war on drugs. In one of the most notorious examples, police in Arizona used traffic stops to enforce federal immigration law.
For Black drivers, pretextual traffic stops—per Jay-Z, “doing 55 in a 54”—are a routine occurrence and the foremost symbol of racial profiling in this country. For many police departments, these violations are used to fill government coffers and prompt devastating cycles of fines, debt, suspended driver’s licenses, and jail time. Black drivers are 20 percent more likely to be stopped, according to a study last year, and almost twice as likely to be searched.
While the article is about speeding, there are numerous additional areas where police work intersects with driving: stops for all sorts of reasons (as noted above), dealing with crashes or road conditions, escorting important people, and police driving the same roads as everyone else in order to address an issue at a particular location.
In many parts of the United States, it would be hard to imagine police without a vehicle or not interacting with vehicles regularly. Even the community policing idea where police spend lots of time in the same community and at the pedestrian level may still require using a vehicle to travel back and forth or to address particular issues they encounter. The sight of police on foot, horse, or bicycle in certain settings may be unusual to many who are used to the cars and flashing lights.
The same kind of methods proposed to limit traffic fatalities (also discussed in this article) or to promote the use of other modes of transportation could also have the effect of reducing the need for police to patrol or drive on roadways. But, reducing the American dependence on or love for driving is a sizable task.