Minarchists do not claim government will "solve all their problems", nor is it clear that government is a "complete disaster". According to the three leading indices of freedom, only 13 nations (out of almost 200) are currently more free than America. America's constitutional republican framework has been by far the most successful in human history. It has been increasing personal and civil liberties almost monotonically for two centuries, and we are among the most economically free nations in the world, with a per-capita GDP exceeded only by Norway and Luxembourg. Our 300 million people live and work in a continent-wide nation with a $13 trillion economy built on a twenty-first century technological infrastructure. By contrast, anarcholibertarians can merely wave toward a couple of medieval island nations with populations and population densities four orders of magnitude less than those of modern industrialized states. As great as America is, we have detailed, redundant, and current empirical evidence backing up the mainstream findings of modern economic science about how market-oriented reforms within the statist framework can make America even more free and even more prosperous. Anarcholibertarians have nothing of the kind to support their moralizing a priori claim that America would be a better place if we completely dismantled our system of rights protection in favor of a promise by liberty-lovers to set a good example of aggression abstinence.
Skip, private markets in rights enforcement is different from private markets in, say, shoes (to use Rod Long's example). They are different because of rivalry and excludability lead to market failures in the provision of one but not the other. Yes, there is such a thing as government failure, but merely citing its existence (as you do above) cannot expunge the idea of market failure from all our economics textbooks. I'm still waiting for you to identify a modern instance of a vacuum in State authority where protection markets have operated not like the Sopranos but rather like anarcholibertarian fantasy. Until you do, I rest on my argument that there is insufficient historical evidence for the proposition that American should try anarchism instead of minarchism. However, I have no problem with anarchists migrating to vacuums of State authority such as in Somalia. Good luck with that.
By the way, we minarchist libertarians don't agree with you that the power to tax should ideally be used to finance whatever is in "the best interest of the majority". Mandatory taxation should fall only on public bads (i.e., monopolizing, consuming, polluting, or congesting the commons) and should finance only public goods (i.e. non-rivalrous non-excludable services for protecting life/liberty/property). Your "best interest of the majority" sounds like a formula for Leviathan.