People with minds broken beyond repair won't change their minds. Either the world will change without them and they will not matter, or they will die off and be replaced with people who do have healthy minds and will change the world.
In any of those cases, trying to reason with a person who has been made mentally ill by societal abuse, is not going to work.
Proof enough that a true Ancap society is practically impossible, simply because the bootstrapping process from where we are to get there would require something very close to a worldwide destruction event to be plausible. And assuming that the NAP is actually in application, the ancaps who would repopulate and educate this planet would be prohibited from
effecting such an outcome. If the event is natural, it is just as likely, perhaps more likely, that ancap social theories would simply die out with those who advocate them in the near term event.
So if you wish to take over the world, you're simply going to have to out-breed the rest; and ancaps aren't known for this skill.