First, I would like to direct you to this post above that Kakugo wrote. Second, I would like to direct you to Libertarian Anarchism: Responses to Ten Objections by Roderick Long.
No system is perfect. There will always be disputes and conflict. Libertarian anarchists do not want the solution to be monopolized and centralized power. We want just the opposite (i.e. decentralized power). There is no way to predict the exact outcomes of anarchy, but there is also no way to predict the exact outcomes of statism. We can point to what would be likely outcomes, but there are no assurances for any system.
Let me provide a different example, the FDA. Drugs require FDA approval, and this takes years. Statist supporters of the FDA like to point out the lives that are saved from requiring FDA approval. Antistate detractors like to point out the deaths that happen because the FDA takes so long to approve drugs. People are going to die either way. But surely letting people decide for themselves if the risk of certain side effects are worth it is the better way.
Apply this to the nuclear arsenal. Something has to happen to the nukes. But what? Well, as established by Kakugo, it is expensive to maintain nuclear weapons. Without a tax base, who is going to be paying to maintain them? Most likely these weapons would be harvested for materials and put to a different use. I cannot prove this. I can only say that it is likely because of the high opportunity cost to maintain the nuclear weapons. And of course, since the nuclear weapons are centralized, all that is required for them to be launched is for the POTUS to give the order. One person has to give the order and then other people will have to carry it out, people who have been conditioned for years to follow orders. Considering the warhawks in the Republican party, do you really want all of that power centralized so that all it takes is one warmongering fool to get elected and start launching?
Or we could decentralize power and sell the nukes off for scrap and be put to other, more productive uses.