How do we reconcile establishing the social kingship of Christ with his saying 'my kingdom is not of this world'?

You have to be careful here, because Augustine grappled with this same question in the City of God. We are not in an ideal situation, and the best arrangement, short of that, is one where the Church holds spiritual power and the secular state holds physical power, and each watches the other as a kind of check and balance. That worked in the Middle Ages: the pope had supreme spiritual power and the king supreme physical power, and each feared the other, the king because he genuinely believed the pope could excommunicate him and leave him on the precipice of hell, and the pope because the king held the physical sword. The Church guided the state morally, holding kings to honesty and justice, and could threaten excommunication if a ruler wronged his people. As late as the 1950s the Church was still the supreme moral voice; if a bishop condemned a film, people would not go see it. That power has largely collapsed today. So the Church is never going to become the outright ruler of politics and money, and it really only functioned well in that medieval symbiosis where neither side was fully supreme, since absolute power corrupts absolutely. We have the kingship of Christ now only in partial form, already but not yet, and the fullness will come only when Christ returns and there is no more competition. In the meantime, the reason the world lasts at all is that the Church is the salt of the earth; the Mass offered somewhere every hour, as Malachi prophesied, appeases the wrath of God so that the world survives one more day, until the last soul is saved and Christ comes on the clouds of glory.