First, my understanding of the Kingship of Christ is basically the same as yours. But the Kingship of Christ is not an SSPX idea. Because SSPX people want the nations to adopt their schism, they are actually urging
against the Kingship of Christ.
Second, the Kingship of Christ is a Catholic idea and is found in the Catechism: CCC 2105 - The duty of offering God genuine worship concerns man both individually and socially. This is the traditional Catholic teaching on the moral duty of individuals and societies toward the true religion and the one Church of Christ. By constantly evangelizing men, the Church works toward enabling them “to infuse the Christian spirit into the mentality and mores, laws and structures of the communities in which [they] live.” The social duty of Christians is to respect and awaken in each man the love of the true and the good. It requires them to make known the worship of the one true religion which subsists in the Catholic and apostolic Church. Christians are called to be the light of the world. Thus, the Church shows forth the kingship of Christ over all creation and in particular over human societies.
source Third, Dignitatis Humanae supports the Kingship of Christ in two ways. The first is by stating in paragraph 1 that the nations have “[a] moral duty…toward the true religion and toward the one Church of Christ.” Thus the Council does not
oppose making Catholicism the State religion, but in fact calls for it.
The second way Dignitatis Humanae supports the social Kingship of Christ is in paragraph 6, where the Council shows that this does not go against the idea of religious liberty: “If, in view of peculiar circumstances obtaining among peoples, special civil recognition is given to one religious community in the constitutional order of society, it is at the same time imperative that the right of all citizens and religious communities to religious freedom should be recognized and made effective in practice.”
I recommend reviewing the Catechism paragraphs 2104-2109, which explain the social Kingship of Christ in more detail:
scborromeo.org/ccc/p3s2c1a1.htm#2104
Let me know if that is helpful. God bless!