Politics is just one aspect of life in the community; to say that Christians should not be involved in politics is to say that we should not participate in the life of the community, but more implies that there is an aspect of community that falls outside the trans formative power of the cross.
It’s not that managing the world and managing the soul are identical but that a mind born again will picture a different kind of society and choose different kinds of fellowship. We might expect a vast commonness between Christian and not Christian thought in the realm of politics being that we are all children of Adam created with common gifts, and so the ways that we try to work out our purposes are largely common to the kind.
The natural law tradition from John Locke to Clarence Thomas is rooted in this recognition of common means applied to common ends. Augustine thought of the city of God and the city of man as over-lapping magisterium with essential elements common to both; a common law pursued toward irreconcilable ends; one toward mere natural glory and the other toward the Glory of God.
Laws embedded in the warp and woof of a well ordered society, that one should not murder, that one should not lie, that one should not steal or covet one’s neighbor’s husband/wife are then not “religion” per se, but mere common sense written in the soul as much as in obvious consequences.
Of course there will be times when both the ends and the means of political life between Christian and not Christian are irreconcilable and at those times Christians will speak truth to power and dissent, possibly facing the gravest of consequences; such is the calling.
In most cases the goods of the Christian life (true and ultimate human flourishing) and the goods of the natural life (self interest and the personal gratification) coincide, if the natural person can be convinced that their true and good interests are served better through reconciling behaviors not with natural instinct but with the alignment of human action with divine design.
Much of contemporary Christian thought veers toward the utter irreconcilability of Christian and not Christian thought, thinking of such as “wholly other” and having no epistemological point of contact, but such has little support in either sacred scripture or Christian tradition.
There is a secondary but related tendency to confuse that which is particular to the Gospel of Jesus with the Law of God, these two things not being the equal. Here we are speaking to a commonness in relation to ethics, or politics as applied ethics, or as to the common grace of a remaining knowledge of good and evil in every person created in the image and likeness of God.
The Gospel is instead the available provision of God for the salvation of the human soul and this is not “common” but peculiar to those who believe the message of Jesus, and so are called “Christian” in regard to their being a follower of Jesus, the Christ of God.