So your ideal society is one in which people get killed by "Professionals with black clothing and silenced weapons" hired by anyone who thinks they are cruel to animals.
It is not ideal society, it is society that may be better than what we have now, or at least not worse, since we have people which get killed by professionals with blue clothing, called on by anyone who thinks they are cruel to animals. Besides, it not necessary to kill. Give very stern warning to these people, maybe with warning shots, and it may be enough to give them notice that they are being watched. This is not so much possible with just calling police who may just come, investigate, and leave if they can't come into house to investigate.
I am opposed to female genital mutilation and I am happy with the present system where people who do that to a girl are named and shamed in court proceedings and their children can be taken into care. Anarchists say that female genital mutilation is a personal moral choice of the parents and that they should be left get on with it.
Yet more example of you confused about anarchy. Anarchy is not every person should be allowed to do whatever he wishes. That is what media portrays anarchy as, making it seem violent, scary, and every man for himself. In anarchy, every person makes their own decision. This also includes decision to name and shame people who do female genital mutilation, and refuse to interact or do business with them. Anarchy is community where everyone is equal.
Do you want everyone to face the risk of mob law? That often misfires. For example, in Wales, a paediatrician was driven out of her home because the locals thought paediatrician is a synonym for paedophile. Or do you want a system of courts whose approval is required for any violent acts and a system of police who are trained to use violence in a minimal way.
Democracy is mob law. If more people believe something is wrong, they will rule that it should be illegal. Look at how much time American mobs spend on voting to support religions, protect children from sex, and making drugs illegal. Everyone is so afraid sex and drugs that their law system is very broken.
And courts can also be private, people can agree to take people to court to stand trial, and private security can be trained to use violence in minimal way. People liked the idea of trials in courts. That is why we have passed laws to agree to such a thing to begin with. So it is likely that people will continue to want private courts too. And it is more costly for private security to commit violence (lost ammunition, damaged property, lost or damaged employees) than for police to engage in violence. Police always get their paycheck no matter what. Even if they kill a child by accident, they just get warning and keep working. Compare how many civilians police in America killed, to how many civilians gangs like Yakuza and Cosa Nostra killed. At least those gangs don't shoot children when they get scared of children with plastic guns.
