My understanding is that the NAP is not compatible with a state because a state insists on a monopoly on violence and you pay taxes regardless of your opinion on on what the money is spent on.
Am I wrong?
That's correct, as the concept of a state is presently known. However, a state that does not insist on such a monopoly on legitimate use of force is not inconceivable. Most people don't know this, but once upon a time the United States was one such nation, and there were
several (but never all) of the many states that did not insist on such a monopoly either. Most of the Western territories (i.e. Utah, Wyoming, etc) didn't have a monopoly on force within those areas as a practical matter, and that is why (despite the relative peacefulness of the age as compared to today) history refers to this period as "The Wild West".
Following it's founding of the nation of Switzerland (of the old confederacy, circa 1291), that nation could also lay credible claim to a national government that didn't insist upon a monopoly on the use of force, as such powers rested in the 'cantons' for centuries, and as a matter of practiality, most of those cantons in turn expected the citizens and clans within to participate in the maintaince of the public order.
Even Great Britain can make
some claim that their societies were not originally dependent upon a government monopoly on the use of legtimate force. For starters, the prison known as "Old Bailey" was traditionally managed by a warden and guard force recruited from within the prison population itself; so the "criminal element" (mostly debt prisoners, really) literally policed their own during their sentences. Also, the British had (at that time) a form of private police force, as anyone could be hired as a 'constable' (as opposed to a sheriff or his deputies) who acted much like how one would expect a private security guard/detective/thug would act. Of couse, local governments could hire constables as well, and that is exactly how modern urban police forces 'evolved' as well as where we get the word "cop", as it's entomology of the word comes from "constable on patrol". Where I live that legacy lives on, as while the Chief of Police (sheriff) is appointed by the mayor, there are two elected positions in each county in Kentucky called 'contables' that impose police powers upon the elected individual, but offers no salary. In practice, the elected constable is just a businessman that runs a private security force and hires off duty or former cops to hang out in banks or hotel lobbies, or serve court summons to civil actions.
So it's not true that we don't have evidence that a state
can't function within the NAP, or that a state can't co-exist with private protection services. It's only in the modern world that most people can no longer imagine such a thing.