No, the libertarian argument is not about the character of the king at all. The argument is that his position -- as one who governs by birthright and cannot be legitimately ousted -- makes him favor rights, legality, peace, and stability. It doesn't mean that bad kings are impossible, it means that even bad kings are nudged in the right direction by the necessity to avoid a revolution, while even good politicians are nudged into the wrong direction by the necessity to get elected.
There is a valid moralistic argument for Christian monarchy as well -- based on the fact that the Church is the repository of the moral teaching and installs the king -- but I did not make that argument in the post you are responding to.
Even bad kings are nudged in the right direction by the necessity to avoid a revolution, while even good politicians are nudged into the wrong direction by the necessity to get elected.
Unfortunately, insecurity of thrones can nudge bad kings to be even worse -- to become all the more oppressive in the fear that their power is threatened. This is a temptation that is less common in republics than in monarchies or dictatorships. For a public official to know that he will be out of office in four years isn't necessarily a bad thing.
A lot depends on the wider cultural context. Comparing monarchy under the best circumstances to representative governments under the worst circumstances is a crooked game.