Great question, very thought provoking. There will be many philosophical (and opposing) viewpoints on this.
What I believe is that there are a number of activities that require large scale infrastructure like roads, water, mass transit, etc. and especially national defense that would not be profitable for private enterprise in many cases (not all, I grant you). Add to this care for the indigent elderly, education, oversight of businesses to prevent fraud & abuse of workers (I grew up in WVa and the pollution, poisoning of the environment, and especially the abuse of miners was horrible - thank goodness for EPA, the Unions & OSHA!), and yes....police protection that would be difficult to provide by private enterprise. I believe your earlier premise was that policing could be privatized but who would protect us if this private police force decides it owns us? Would they strong-arm those that don't want to pay? At least in today's environment there is the Federal Justice department that forces change upon local jurisdictions when flagrantly needed. Too slow, not as broad as we'd like but it is there. I actually see privatization of local police as even more dangerous than the current situation.
Police do not have to be the enemy. There was a terrific article on one of the news shows recently about the completely different approach being taken in Watts. It is a return to the Andy Griffith way of policing, and is working wonders on the crime rate and the relationship between police and the community. They befriend, they counsel, and only arrest as a last resort. They don't have quotas, their goal is reduced crime not total arrests or even solve rates. Sorry I don't remember which show, I was medicated, lol! If we could somehow get all the police in this country to follow their lead, it would move us a long way toward what I think you are aiming for.