What that person is saying makes no sense. Organised aggression is a pretty big claim right from the beginning, but the "zero-government" crowd are never able to properly back it up. They just point to examples of corruption, e.g.: police brutality in their particular society, and they assume it must be the same everywhere else.
Who said anything about corruption? Anyone who buys things with their own money and uses it in the privacy of their own home is at risk of such oppression. Anyone who has a relationship not sanctioned by the government, who refuses to pay taxes to keep posessions of the person they were with after their death would face such an oppression.
Another big claim, and it sounds like a straw man to me. Who believes that a human being is righteously authorised to murder, brutalise, or cage another human being for peacefully resisting an order?
Anyone who lives under the rule of police, FBI, Interpol, MI6, SSF, or any military organization of another country. Wall Street occupiers have demostrated vividly that peaceful protests often result in you finding your hands secured with zipties, and yourself being carted off to a paddy waggon. People in other more opressive countries find themselves arrested and carted away for things like speaking out against the government, distributing information governments don't particularly like (Wikileaks), or peacefully enjoying things that they don't think the government should make illegal (alcohol used to be on that list).
The problem arises when a society decides for others that it "owns" them, and decides for others that they must follow its laws, even if that other has never agreed to it. Another problem is that, if laws were simply "common sense" things that were easy to follow and understand, I would agree with you. However, laws now require multiple thousand page volumes to list, and in many places "ignorance of the law is not a defense." So, unless you carry around law books to check your every action, the laws have become nothing but a way for the government to have an advantage over you. The situation has quite literally turned into one where everyone is probably breaking the law in some way, and it is simply up to the government to decide who should be punished. this is how totalitarian states operate.
A much more blatantly vivid example is this is how the Church during the Dark Ages operated. Sex, the most basic of human instincts and desires, was largely illegal. Not because it was unethical, or harmed the participants, but only because the church knew it was something everyone wanted and that everyone was doing it, and having such a law on the books allowed it to selectively pick and choose which undesirable characters it wished to send to the dungeons.
You don't think that taking out those who commit criminal acts out of society, or even a threat of retaliation or retribution, would minimize cruelty and harm in a society?
A thief breaks into someone's house and attempts to steal some stuff, but is caught red-handed. In a society with a State, there is certainly some kind of due process that ought to be followed. E.g.: the home owner calls the police, the police catch the thief, arrest him, lock him in a cell, and depending on the severity of the offence he might be remanded in custody until a court hearing, remanded at large until a court hearing, or maybe just discharged the next day. However, with no laws apart from the N.A.P., the home owner would grant himself "judge, jury, and executioner" rights, and commit some act of violence against the intruder because none of the intruder's rights have been suspended, forfeited, or even questioned. If the intruder maintains all of his/her "natural rights" at all times, then any act against them would indeed be a violation and cruel.
If you want a more precise example for AnCap, expand your example beyond borders. If you cross the border into China, Pakistan, or Saudi Arabia, and commit a crime there, you WILL be imprisoned, and likely killed, regardless of what your society thinks your rights out to be. That's even the case in some states and provinces within the countries you list, where it is legal to shoot your intruder. So, there really is no difference between the statist world you feel comfortable in, and the scary AnCap you are imagining.