'Splain. There is nothing any more "sacramental" about placing a crown than taking an oath and proclaiming "so help me God."
From "Coronations in Catholic Theology":
Whence, indeed, came the authority of Kingship itself? True it was, that in most countries, the Crown was passed along by hereditary right...But in both sets of cases, the added charism, so to speak, required something more...The anointing of holy oil...took place at the rite of coronation. While the coronation was not itself held generally to confer the Kingship, it nevertheless seemed to be necessary for the royal personage to enjoy the fullness of the graces thereof.
From The Lord of the Rings: A Catholic View:
...it will be necessary to describe a little of the uniquely Catholic world view. In fine, it is a sacramental one. At the heart of all Catholic life is a miracle, a mystery, the Blessed Sacrament. Surrounded traditionally by ritual and awe, it has been the formative aspect of Catholic art, drama, and poetry. The coronation of Kings, swearing of oaths, marriages, celebrations of feast-days, all have a Eucharistic character."
Read both articles; they're worth it.
I would add that the existence of a monarchy, with a religious coronation ceremony marking the beginning of each sovereign's reign, allows the Church to shape the character of the State in a way that is not really possible in republics. By anointing the king or queen, the archbishop confers a type of grace upon the living icon of the nation while making it clear that king is still subordinant to God. Sadly, Europe's surviving continental monarchies have abolished their coronation ceremonies, replacing them with secular "inaugurations" similar to those of presidents. Only the (Anglican) British monarchy, ironically, retains what was originally a thoroughly Catholic ceremony. Did you know that the rite with which Queen Elizabeth II was crowned in 1953 was essentially the same service used for King Edgar in 973, almost a thousand years earlier? I find that incredibly inspiring, moving, and comforting. What can republics offer to compare to that kind of continuity?
From The Rulers of Britain: "Edgar waited to be officially crowned until he was thirty--the age when a man might be ordained a priest. The ceremony emphasized the sacred nature of kingship, and of the monarch as Christ's deputy."