I can think of two main errors made about the relationship between the gospel and politics. Each of them has multiple instantiations:
In this case, Christians see that the gospel has specific political implications and then associate those implications with the gospel itself.
In this case, Christians see that the gospel is central and supreme and therefore ignore domains, ideas, and policies not central to the gospel.
Both of these happen on the theological and political right and left.
I think the relative importance of politics, in comparison to the gospel, does make non-participation necessary for some people (like some had to sell all their possessions in the gospels). Similarly, I think that the fact that there are right and wrong political positions, or at least right or wrong political aims means that Christians, generally, ought to care about politics to love their neighbor and see to the well-being of their children and grandchildren.
But I think that it is wrong to elevate politics (particularly as understood in American civic life) to the center as a primary matter of discipleship. For instance, one can be a Christian with little to no understanding of what the Old Testament is for (see Romans 14). Understanding one’s local political system and how best to maneuver it for maximal flourishing and minimal corruption is a labyrinth far more complicated and far less central to the life of the individual Christian.