For your consideration:
…However, one sword ought to be subordinated to the other and temporal authority, subjected to spiritual power. For since the Apostle said: 'There is no power except from God and the things that are, are ordained of God' [Rom 13:1-2], but they would not be ordained if one sword were not subordinated to the other and if the inferior one, as it were, were not led upwards by the other.
For, according to the Blessed Dionysius, it is a law of the divinity that the lowest things reach the highest place by intermediaries. Then, according to the order of the universe, all things are not led back to order equally and immediately, but the lowest by the intermediary, and the inferior by the superior. Hence we must recognize the more clearly that spiritual power surpasses in dignity and in nobility any temporal power whatever, as spiritual things surpass the temporal. This we see very clearly also by the payment, benediction, and consecration of the tithes, but the acceptance of power itself and by the government even of things. For with truth as our witness, it belongs to spiritual power to establish the terrestrial power and to pass judgement if it has not been good. Thus is accomplished the prophecy of Jeremias concerning the Church and the ecclesiastical power: 'Behold to-day I have placed you over nations, and over kingdoms' and the rest.
Boniface VIII, Unam Sanctam (One God, One Faith, One Spiritual Authority) November 18, 1302
This says close to the same thing you're asserting, but showing that there is a more direct link.
Bellarmine was the first to clarify the idea that the Pope does not have direct authority over a temporal sovereign. The Pope at the time did not like that idea at all, however it later became the standard Church thinking. John Locke and Algernon Sidney got on board and furthered Bellarmine's understanding that the king does not have a divine right to rule. Therefore, the king has no authority to interfere in things spiritual.
The easiest way to think about this is the authority within a single church in America. The church is sovereign in things spiritual, but is indirectly submissive to temporal authority. For example, when a church builds its building, it must abide by the temporal-based fire codes. Now the temporal authority has no direct authority over the spiritual aspect of the church as in preaching or helping people spiritually, but it can tell the church that its building is not with the fire code. The temporal realm has direct authority over the people when excersizing its temporal authority, like enforcing laws that relate to the temporal realm.
Now what our Founders did was to make the concrete distinction between the spiritual and temporal realms by law and influence where each works directly within its area and indirectly when it deals with the other. Their intent was to further the moral fabric of the people, by making both realms responsible for encouraging good moral behavior, one directing spiritually and the other by providing for good laws which enforce the temporal action.