Any 21st century monarchist movement must come to terms with pre-existing command structures which manage conflicts within Western civil society -- the rise of mass man, increasing complexity and atomization, resource scarcity and overpopulation, the polarization unleashed by ideological warfare, economic shifts and demographic change, and so on. These command structures are defined by what Burnham identified as managerialism, a form of political machinery which operates as a bureaucratic coalition of experts and elites, and they are extremely effective. In Europe, the EU and what is euphemistically termed the 'international community' are responsible for maintaining this machinery.
As fascists and Marxists alike recognized during the interwar period, monarchy is too sluggish, too weighed down by obsolete traditions to respond to emerging historical developments. But eventually they, too, found themselves on the wrong side of history, leaving managerialism as the only form of government legitimately equipped to handle the vicissitudes of modernity.
Installing royal figureheads won't change anything in this regard. The only two solutions are to either overturn the entire system, like Lamar in Walker Percy's Lancelot, or else usurp the managers themselves and change the system from the inside out (Mencius Moldbug's solution). What this resurgence of monarchism signifies is not an imminent restoration of the throne, but something equally encouraging: dissatisfaction with the present liberal status quo. As for what comes next, nobody can say, but I think it's obvious that it will fulfill that famous Chinese curse: "may you live in interesting times"!