Show me where in the 1st Amendment it says that there is "separation of church and state". It only protects religion from the government, not the government from religion.
Same energy as "we're not a democracy, we're a republic".
True. There is no question in the practical meaning of the word republic, the word now simply means that there is no monarch even if a figurehead. Example: Pinochet's Chile was a republic in that sense. Apartheid-era South Africa was a republic. Iran is a republic. All Commie states were or are republics.
So if the cost of maintaining a liberal society should be the maintenance of a constitutional monarch, then I will put up with the constitutional monarch. Plenty of tyrants have cheapened the meaning of the word republic to include some regimes that rival that of Ivan the Terrible for sheer horror.