Lets start with privately funded security.
What happens when one such security provider becomes tyrannical and starts abusing people?
What happens when one such security provider becomes tyrannical and starts abusing people?
Fallacy: begging the question.
You make an assumption that security is only provided by privately funded security businesses and then you make a conclusion that we ancaps should now argue against. Well many ancaps may fall into this trap, but I wont. I never conceded to your assumption that private security firms would be the only way security would be provided in an ancap society. Quite the opposite, I strongly think that in an ancap society people would generally realize that ultimately they themselves bare the responsibility for their security and that means they themselves would need to find ways of providing it.
How does this change your argument? Well it changes it a lot. Originally your assumption (which I did not concede to) assumes that people have no choice, or better said no other means to protect themselves from a rouge agency that they hired to provide for their security. The truth however is much more likely that people would be highly capable of quickly dealing with such an agency and that even the threat of such a swift defense would be enough of a deterrent for such an agency to never even attempting it.
And here's a broader point you have to understand about ancap theory. We ancaps usually, if we are honest, do NOT have almost any answers as to how certain problems in such a society would get solved. Why? Because the solutions could only ever come from a market regulated strictly by consumption i.e. a free market and not any single person. Just like no person 200 years ago could have given a correct or even an answer in the right neighborhood when asked how the fields would be worked on and food produced if slavery was abolished.
But not having any answers is irrelevant. What are relevant are the foundational principles upon which a society is structured. It didn't matter that no one could have given the answer that "big metal machines with many consecutive tiny explosions of petroleum inside of them" would work the fields because all that was important was that if you want to live in a society that will offer you a good life, slavery couldn't be a principle upon which it was built.
And this will be pretty much the same answer of an honest ancap to any of your "issues" you might raise of how an ancap society might solve certain problems: "We don't know, but it's also irrelevant that we don't. Our theory is valid because of the principles not because of the solutions any one of us might be able to imagine."