I have thought a great deal about this. It has been a loooong haul for me, a libertarian and a Libertarian, to reach this position re: monarchy versus democracy.
I am nott finally there, just getting closer in my thinking.
As of today, if asked how I wood envision an 'ideal' monarchy, it wood be governed by a royal family, nott just one supreme monarch, and it wood do so in consultation with a democratically elected parliament to provide feedback and guidance to the royal family.
Perhaps the power would be held by a panel of royal family members - as the whole family has a stake in the future of the kingdom and this helps to buffer the 'Mad King' that periodically will arise. The royal family can develop its own managerial and governing panel and decision-making processes, subject to 'guidance' from the parliament.
The objective is to place a sovereign with authority over the parliament so as to restrain democracy from its inherent short-termism and tendency for Madness of the Masses and fiscal/monetary self-immolation.