If you accept the criticism it makes of democracy you are already basically Neoreactionary. Only about half of them advocate monarchy as what should replace our current order, remember no one said the journalist did an excellent job reporting about us. While I can't even speak for those who do advocate Monarchy, only for myself, here some of my reasons for finding it well worth investigating and advocating:

1. Good enough - You need not think it an ideal form of government, but if you look at it and conclude it is better than democracy and nearly anything else tried from time to time so far, why not advocate for it? We know it can be done with humans and can be stable. This is not the case with some of the proposed theoretical forms of government. Social engineering is dangerous, you want fail safes. If you want to be careful and small c-conservative it is hard to do better than monarchy, it is as old as civilization, an institution that can create bronze age empires or transform a feudal society into an industrial one.

2. Simplicity - Of the proposed other proposed alternative forms of governments it is the one most easily accurately explained to nearly anyone. Simplicity and emotional resonance are important features with many consequences. For example when Moldbuggians say society would benefit from formalization they should aim for a bare bones OS for this to be feasible. Formalization is the process where the gap between the actual and claimed functioning of a social institution is closed as much as possible in order to reduce disinformation. This is considered good because uncertainty results in politics/war. There are also costs for keeping people in positions of responsibility sane and not accidentally ending up believing in disinformation if such is common around them. Not bothering to keep them sane at all seems bad.

3. Agile experimentation - Social experimentation is however useful, especially since the same solutions won't work for all kinds of societies in all situations. It is a system that can be easily adjusted for either robustness or flexibility as needed. A monarch has simple logistics to set up or allow social experiments. Futarchy, Neocameralism... why risk running a society on this OS rather than set up a more robust one and then test it within its confines? East India Companies, Free Cities, religious orders are common in the history of Western monarchy. Indeed you can look at Constitutional Monarchy in modern democratic countries as experiments that where either judged successful or an experiment that breached containment. Even in this case of breach the form of monarchy was still preserved however and might possibly be revived at a future point in time.

4. Responsible ideology crafting - Many Neoreactionaries think the relative political stability of the Western world of the past 70 years will not last. Historically transition from some kind of republic to military dictatorship is common. Rule by leader of victorious conquering army, has historically show successful transition to monarchy, as all dynasties where basically founded by them. Even if in itself such a change isn't likely in the West, the unlikely situations where neoreactionary criticism of democracy would be taken seriously and guide policy, is one where the most likely victor of the social instability is not an ideal representation of a Neoreactionary CEO philosopher but a military dictator. We should try and plan social reform constrained by logistics of the likeliest outcome of our ideas becoming important, otherwise we are irresponsible. Indeed this might have been the grand crime of Communist theorists.

5. Low Hanging Fruit - It has been understudied by modern intellectuals who furthermore are biased against it. Compare how much modern theoretical work has been done on Democracy vs. Monarchy. See number wikipedia articles for a quick proxy. This is perhaps practical given the situation we find ourselves in but also somewhat absurd. For example as far as I'm aware no one outside reaction has in depth considered the ridiculously obvious idea of King as Schelling Point! Modern game theory, cognitive science and even sociology unleashed on studying monarchy would reveal treasures, even if we eventually decide we don't want to implement it.