For some like John Stackhouse, keeping the church out of politics is a big duh (via the juicy ecumenists):
10. Because no one trained you properly to get involved with politics—and a little seminar, however exciting, won’t make up for that yawning deficit. (Do you think politicians can be trained to be pastors by attending a seminar?)
9. Because no one hired you to get involved with politics. (And if they did, they shouldn’t have: See #10.)
8. Because pastors are supposed to call us toward the ideal and the ultimate, while politicians have to compromise over the real and the immediate.
7. Because the Scriptures (your main area of intellectual expertise—right?) are, at best, only suggestive and regulative over the field of politics (a quite different area of intellectual expertise—right? See #10 again).
6. Because you’ll alienate a considerable part of your constituency who see political matters differently, and will hold that difference against you, thus losing the benefits of your pastoral care and authority.
5. Because you need to consider the troubling fact that you’re not alienating a considerable part of your constituency, so why is your church so uniform in its politics?
4. Because governments come and go, and you need to reserve the sacred right to prophesy to whoever is in power.
3. Because politicians come and go, and you need to reserve the sacred right to comfort whoever is not, or no longer, in power.
2. Because politics brings out the worst in people, and you’re supposed to bring out the best in people.
1. Because politics brings out the worst in people, and unless you’re an exception (like Tommy Douglas), politics will bring out the worst in you.
But for others, the world would be a better place if the church were “running things”:
The fate of the world in every epoch since the Incarnation has been bound up with the state of the Church. The Church’s power to renew the face of the earth involves not only a restoration of faith, hope and charity in the souls of men, but also the defense of natural reason against the onslaughts of sophists in every age. She alone has upheld the correct synthesis of fides et ratio. The Church’s success in accomplishing this mighty work throughout history has always depended upon her vigor in advancing what she calls the Social Kingship of Christ. But it is precisely Christ’s social reign that the “modern world” has rejected, while churchmen fall silent regarding the claims of Christ the King on men and nations. Today, she not only retreats from any confrontation with “the rulers of the world of this darkness” and “the spirits of wickedness in the high places,” but seeks obsessively to dialogue and collaborate with the very forces that desire nothing more ardently than the Church’s final surrender to the spirit of the age.
And yet the truth remains. As Pius X insisted at the turn of the 20th century: “Society cannot be set up unless the Church lays the foundations and supervises the work; no, civilization is not something yet to be found, nor is the New City to be built on hazy notions; it has been in existence and still is: it is Christian civilization, it is the Catholic City. It has only to be set up and restored continually against the unremitting attacks of insane dreamers, rebels and miscreants. OMNIA INSTAURARE IN CHRISTO.”
. . .
The ecclesial crisis and the intimately related civilizational crisis will end only when the Church’s offer of social metanoia is renewed once again. But only the Vicar of Christ can effectively extend that offer to the world. Only he can end what amounts to an unprecedented de facto suspension of the Church’s true mission in the name of a Council whose restless “spirit,” moving far beyond even the problematical conciliar texts, has produced what Benedict XVI, speaking just days before his mysterious abdication of the papacy, described as “so many disasters, so many problems, so much suffering” in the Church.
Therefore, we implore the reigning Roman Pontiff to reverse the Church’s course of the past fifty years, abandoning a disastrous “opening to the world” and an endless “dialogue” and fruitless collaboration with the Church’s implacable opponents. With respect to the Synod, we urgently petition the Pope to put a stop to all further efforts to use the synodal process to undermine the indissolubility of marriage—and thus the entire moral edifice of the Church—by means of a sophistical disjunction between doctrine and practice, making a mockery not only of the words of Our Lord Himself but also of the teaching of John Paul II that “only by the acceptance of the Gospel are the hopes that man legitimately places in marriage and in the family capable of being fulfilled.” 
Which makes me wonder yet again if theonomy, neo-Calvinism, and Covenanting (Scottish style) are gateway drugs to Roman Catholicism. (But it does show how much Roman Catholicism has changed since Vatican II.)