There are still 12 monarchies in Europe, some of them Catholic, some Protestant. Your theory does not work there. The monarchies and Catholicism itself evolved (to a great degree, courtesy of the French Revolution), while Islam seems to have stalled.
The job of the monarchy today is largely traditional. They pass and bless today's (better) government systems.
> The monarchies and Catholicism itself evolved, while Islam seems to have stalled.
Do you know the story of Iranian Democracy from 1960s?
There are... complications... to democracies in the middle east.
I wish: <https://www.theguardian.com/uk-news/2021/feb/08/royals-vette...>.
> There are... complications... to democracies in the middle east.
It seems there are complications to all forms of government in the Middle East.
Yes and no...
Firstly there's not been a King of England since 1707, he's king of the United Kingdom (and fourteen other places) and secondly it really depends who you ask. On paper sovereignty flows from the 'Crown in Parliament', many powers are actually the King's and are now vested in other parts of the state. Since the 19th century these powers tend to be used 'on the advice of the Prime Minister / Cabinet'. In theory the King could go against this advice, but Parliament ritually reminds [1] the King's representative what happened the last time a king got too uppity and the courts can still rule on the limits of these powers. There's one view which says the people are fundamentally sovereign (very relevant during Charles I's trial for treason), but while this might be true in practice the orthodox view is that sovereignty comes from the 'King-in-Parliament', a composite body of the King, House of Commons, and House of Lords acting in concert.
The point is that when Andy Burnham exercises his power in many cases he's in a sense doing 'su king' by virtue of his role as PM rather than acting in his own right. The PM role is weird as well, traditionally (and in my opinion correctly) the PM is merely the first among equals in a collectively responsible Cabinet, the term 'Prime Minister' actually started life as an insult. Of course since the late 20th century the role has become increasingly presidential, but this seems true in Westminster-style systems generally.
[1] https://en.wikipedia.org/wiki/Black_Rod#State_Opening_of_Par...