1> Patrol public spaces to deter illegal behavior in those places, direct traffic, punish traffic offenders, offer directions to tourists, etc
2> Respond to calls from citizens, investigate the crimes that those calls are about, locate and apprehend the associated criminals.
I've only lived in places sparsely populated enough that <1> is mostly impractical except along highways and around major construction sites.
The VAST majority of the value provided to people by the police clearly comes from <2> - it's the reason that no sane person would mug me (or kill me) for $40 cash on an unpatrolled country road, or invade my home and take up residence there against my will. 99.9% of citizens benefit from <2>, as it's the main deterrent to any antisocial person coming and taking whatever they have of value.
It seems to me (though I don't have data) that most (nearly all?) the mistreatment of (maybe mainly black) citizens by police that has been garnering media attention over the past decade or so, happens during the course of <1>.
Plainly, I don't believe in deterrence as effective or a substantial reason why most people don't engage in bad behavior.
60% of murders go unsolved because a sane murderer does not commit a murder that would be easy to solve. I'm sure many tempting, high-reward murders do not happen because they can't be committed without an ensuing police investigation.
Suppose 1% of people are currently criminals (I don't know the real number). Without an authority investigating and punishing crimes, that number might increase to 1.1% or even 2%. But a small fraction of those would likely increase their criminal behavior by an order of magnitude.