The reason we have a civilian policing body is because of a human behavior called "specialization".
You seem (and please correct me if I'm misreading you) to be vaguely suggesting some sort of mass societal approach to policing where people take care of each other communally as the need arises - a very Neo-Anarchist approach to things.
But specialization exists for a reason, particularly in fields that require a high degree of training and expertise, and matters of law are just such a field. There are very good reasons we have dedicated professionals serving to operate our justice system.
We have lawyers and courts because laws are complex and exacting, and we cannot rely upon nor reasonably expect the average citizen to be adequately aware of their legal status and rights much of the time. We have law enforcement officers for much the same reason - but also because the enforcement of our laws requires some very complicated or difficult endeavors.
Criminal investigations, for example, would be a nightmare to conduct communally - both logistically, and socially. It's just such a delicate and complicated task that it requires trained specialists to perform correctly and fairly.
I would personally be horrified of communal criminal investigations, as I have almost complete certainty they would devolve into "mob justice" immediately. It's just something I don't want being handled by anyone who isn't an expert.
Yet even more significant than the mere necessity of specialization in these sorts of endeavors, I find that your key objections to the extant police force are in no way innate to the system or otherwise insurmountable.
Most workplaces operate on their own unique sets of rules - but particularly government jobs do. The police are not the only ones legally empowered to perform certain tasks ordinarily not trusted to individual citizens. You can't argue against the police on the grounds that they are granted special allowances without arguing against every other office that is granted special allowances.
On the other hand, if you only take issue with certain of the allowances made, or feel certain of the powers granted are too far reaching, those things can be changed without abolishing the entire system.
You claim that the police are separate and distinct. It's arguable how true that actually is, but if you feel that things need to be less so, that's a change we can seek to make and a reform we can enact. The police are not intrinsically removed from other civilians, and it is perfectly possible and reasonable to hold them accountable in the same manner as everyone else.
It really all depends on the heart of your objection.
If you object to the granting of special powers to a certain portion of the population, I fear you have no hope of satisfaction. Specialization isn't going away any time soon, and special powers granted to certain individuals placed in the public trust in order to perform essential duties beyond the scope of ordinary citizens is practically the cornerstone of human civilization.
On the other hand, if you object only to the specific powers and privileges, on the grounds that they take matters too far or are in various ways unreasonable or unjust, then you're in luck - those details can be changed, and the system updated and improved.
Reform is overwhelmingly almost always possible, and people are far more willing to fix a problem with an extant structure than they are with trying to rebuild the entire ediface from the ground up.