The natural tendency is for Church and Sovereign to become one (interpreting Religion and Church broadly to include progressivism as a religion and Harvard as its Church) If one, the question goes away.
But sometimes church and sovereign are geographically different, as when the Holy Roman Empire lost power, resulting in one Roman Catholic Church and many Roman Catholic Kings. Whereupon trouble ensues, and the question becomes urgent. What tends to happen is that the Pope proclaims himself superior to Kings, but is under the thumb of one particular King, so that the supremacy of the Pope started to look suspiciously similar to the supremacy of the Habsburgs. Today we see that Harvard has alarmingly great power in Iran, and Obama alarmingly little, and we see state department functionaries taking power in supposedly independent states, which state department functionaries were usually educated within a very short distance of Harvard.
The solution to this problem is given by the Chrismation of Solomon: Zadok the Priest & Nathan the Prophet Formally Chrismated Solomon King
Formally, Solomon became King because the priest and prophet anointed him so, making church authority supreme over sovereign authority. (Chrismation is Eastern Orthodox Christian language for anointing with oil. Supposedly the oil goes back to biblical times, having been continually diluted with fresh oil.)
In actual substance, Solomon became King by murdering his brother Adonijah, arguably the legitimate heir, in a fight over Abishag, the most beautiful woman in Israel, even though it was illegal and immoral for either of them to possess her, and even though there is no mention in the bible that she intentionally did anything to tempt either of them, and by shedding the innocent blood of Joab in the tabernacle, thereby desecrating the tabernacle.
However by formally submitting to the Church, the sovereign pointed away from the unpleasantness of a messy succession, and to God. In substance, Solomon was in charge, largely due to craftiness and ruthlessness. In form, the church was superior to the state. In substance, Solomon dismissed one priest and appointed another, violating the principle of hereditary priestly succession. 1 Kings 1 and 1 Kings 2 lists numerous morally dubious or openly wicked killings and purges by Solomon over the succession question, and 1 Kings 2 concludes:
So the king commanded Benaiah the son of Jehoiada; which went out, and fell upon him, that he died. And the kingdom was established in the hand of Solomon.
So was it blood or oil that made Solomon King?
Thus the bible points to the formal authority of Church over Sovereign, in order to give the sovereign authority, in order to make Kingship inspiring rather than demoralizing, but the substance of authority belongs to the sovereign, not the the Church. Napoleon was wrong to crown himself, not wrong to have himself crowned.
If there is a distinction between Church and State, and there usually is not, nor should there be, we should take the Chrismation of Solomon to demonstrate the proper relationship of Church and State. The Church should be formally superior, to make Kingship holy, to give dignity and virtue to the state. But actual superiority leads to the problems encountered with the sons of Samuel, and with Papal indulgences, and with Habsburg empire, and with Harvard.