The advantage of monarchy is that the ruler does not normally face a rival who holds an equal right to rule. That makes him the opposite of a modern dictator who rules because the army followed him, and will be deposed by the next person who can get the army to follow him instead.
Her Majesty, for all her strengths, is I fear a little past the stage of being able to lead a counter-revolution. She could of course serve as a figurehead, but that would not fill the need for a stable government that could concentrate on governing well rather than retaining power. The shogun with real power would have rivals who were his nominal equals, and struggles among them would dominate government just as her nominal subordinates do today in Whitehall.
No, we need a ruler with real personal power, who cannot be replaced by a rival with equal legitimacy. If the current sovereign is too old, we must wait for her successor. Unfortunately, the idea of the Prince of Wales assuming personal power as an autocrat comes over as somewhat ludicrous. Even if I am misjudging him, and he does have the inclination and competence to rule, it seems far-fetched that he could command the organs of state to support him.
As an aside, I might seem to be contradicting myself here - my point is that we need a ruler not responsible to popular opinion, yet I am ruling out King Charles III on the ground that the army etc. will not wear it. However, the value of a monarchy is that once established, its legitimacy is inherent and not dependent on external opinion polls or power struggles. To go from a failed democracy to a monarchy nevertheless will necessarily entail a power struggle of some kind - the first monarch will have a harder job than his successors.
Leaving Charles aside, then we look at the next generation. There, possibly through mere ignorance of their actual nature, we have grounds for hope. The long-cherished links between the Royal Family and the military are much stronger than Charles managed, the young men show admirable willingness to defy popular and fashionable opinion, and not in the direction of exotic mysticism or deep environmentalism either. The next decade will very likely crush my hopes, but based on what we know now it remains at least conceivable that in the kind of degringolade which hangs over this century, an emergency seizing of power in royal hands could be the response to one disaster or another.
It is a long shot, but it is more imaginable than any other route to non-political government that I can think of.
Of course, monarchy does not eliminate all the problems of politics. The fact that the monarch has an extra legitimacy from who he is gives him a head start in holding off rivals who cannot duplicate his every other asset, but there is a level of political lead which will overcome that advantage. The King therefore needs to do expend that much less effort on security. That itself makes his government more pleasant than a dictator's would be, which reinforces both his own position and the legitimacy of monarchy as a concept. But if he is unwise, he will fall as many past monarchs have.
Not that the monarch needs to be exceptional. His absolute power over government does not mean he has to make every decision. He merely has to hire expert managers. The essential feature is that the hired manager is not a public figure, and does not command any legitimacy of his own - he can be fired at will by the monarch. The King must take care that the manager is doing well, but that is a far easier job than the management itself.
There are also the other difficulties inherent in monarchy, which arise when the succession is unclear, or when the rightful monarch is incapacitated. It would take a new Wars of the Roses, though, to bring the quality of royal government down to the level of recent Prime Ministers.