The Police are an essential service in any human civilisation, without which crimes would go entirely unpunished, criminals could do anything they want with impunity, there would be no law and safety, because the law is only as strong as those who enforce it. Society would collapse into chaos.

Not true - formal policing forces are a very recent development in Western civilization.

At best you’d have random disconnected groups of civilians banding together to enforce their own moral “law” on those they deem criminals, but they are unaccountable, unregulated, and no more than a lynching mob, even with good intentions.

More like citizens' arrests to bring criminals to court.

The issue is, of course, that now that there are people with no training attempting to subdue each other, with no indication of the legitimacy of the struggle or potential to 'impose' a socially acceptable conflict resolution if the arrest is disputed. I don't know that "Everyone is now a cop" is really better so much as a recipe for a glorified gang war leading to a small group of enforcers without any institutional oversights.

Wonder what we should call them, for short?

I do agree that generally policing in society is a net good, as vigorously as American police attempt to drive that into the red and as vigorously as politicians try to push policing as a curative instead of an alleviation of symptoms. The US system really needs to be torn down completely, and most cops disqualified from any future policing duties. The siege mentality is so ingrained in so many departments that I can't imagine them being able to reintegrate into any modern policing system, much less a rationalized socialist policing system.