Some Christians don't get involved in politics and leave the material world concerns to others.
But then on the other side there are those who let their religion inform and guide their political choices.
Then again, there are those who really take advantage of people's religious beliefs to bolster their own power... generating lots of religious sounding language, but for their own purposes.
* Would it be better to steer clear of politics and politicians and rely completely on Jesus?
From what I've read of the gospels, that seems more in line with what Jesus was saying.
If we pay attention to the Gospels, it's pretty hard to come away thinking Jesus had nothing to say politically. The catch is that the "politics" of the Gospels is radically different than what we usually consider politics.
Jesus went around preaching about a kingdom. That this kingdom isn't like the kingdoms and powers of this present fallen age doesn't change the fact that it is called "kingdom".
The Cross was a political act, Jesus died an enemy of the state. That is, again, political. But, again, the Cross subverts the political. What Rome used as a tool of terrorism and subjugation, Jesus transforms into the instrument of peace that overcomes and defies the powers of this fallen world. So St. Paul says that Christ made "a public spectacle" of the powers by the means of His cross.
Edited to add:
And then there is the Gospel itself. This word "gospel", "good news", had significance and meaning in the ancient Greco-Roman world. From the time of ancient Greece right into the midst of the Roman empire this word was used to describe the proclaiming major events. When the Persians were defeated when they tried to conquer Greece, heralds went out to preach the "gospel" of the victory over the Persian forces to all the towns and people.
In Anatolia there was this inscription on a building in Priene,
"'Since Providence, which has ordered all things and is deeply interested in our life, has set in most perfect order by giving us Augustus, whom she filled with virtue that he might benefit humankind, sending him as a savior, both for us and for our descendants, that he might end war and arrange all things, and since he, Caesar, by his appearance (excelled even our anticipations), surpassing all previous benefactors, and not even leaving to posterity any hope of surpassing what he has done, and since the birthday of the god Augustus was the beginning of the gospel for the world that came by reason of him,'"
A gospel was a proclamation of victory, of major achievement, of the ascension of a king.
And so, then, we read this in the Gospel of Mark,
"Now after John was arrested, Jesus came into Galilee, proclaiming the gospel of God, and saying, 'The time is fulfilled, and the kingdom of God is at hand; repent and believe the gospel.'" - Mark 1:14-15
The Gospel of Jesus Christ is, in a sense, a direct challenge to all other gospels. Such as Caesar's gospel.
-CryptoLutheran
