Mike we won't convince each other. I will not try, and it is as much out of respect and love as it is out of knowing there is no use.
I challenge your notions. For example, "where there is an absence of God in any arena of life and culture evil is present." Objectively, God is not absent. Objectively, evil exists where men do not like to retain God in their thoughts. The evil begins inwardly and comes from the darkened heart. Therefore, evil in "arenas of life and culture" exist not because of ungodly people in government, but because ungodly people are ungodly. There is actually unfolding before us proof positive that our efforts as Christians to weight the dice of government in "our" direction has only corrupted Christian faith (promoting ungodliness) and not proof that godly men and women in government (assuming without proof there are any) preserve moral health or prevent moral death. This is easily seen in mere observation without any particular appeal to the Scriptures which set forth a singular theme that civil government is the enemy of Christ.
Yes, we need God utterly. But, I also would ask why you fragment life into these arenas. God does not. I suggest to you, and the thinking of others that think as you do, that this fragmentation is a convenience to allow you to make your point rhetorically without actually setting forth substantive truth. I doubt you see it and I know you will disagree. It is nearly impossible to "get" someone to see that fragmentation of life is antithetical to Scripture.
Fragmentation in this way seems innocuous but it is devastating. It is an American modern invention of thought and it is thoroughly bureaucratic. It is flatly not Scriptural. The very thing you use as an identifier of our need for God is itself evidence for our need of Him in precisely the opposite way, not in arenas but fully. Men who come fully to Christ and are fully converted, fully convinced of our own badness and of His righteousness see nothing of a need for better human government, but see only a full need for Jesus to return and subject all things to Himself, knowing that we cannot do it for Him by our own devices, including and particularly the device of government.
"Christian leaders" are to be preferred? Christian leaders, in your proposition, would be "better" pimps than unsaved men. They would be better drug dealers and mafiosos. If only Christians would enter prostitution, drug cartels and criminal enterprises more, those areas could be "better." Or, do you think honestly that government is qualitatively different from those things in the face of the length of governmental history from 1 Samuel 8 until now? A history that, by the way, Samuel explained would unfold from the time that Israel rejected God's exclusive rule and chose for a co-opted reign of God and man. An explanation that defined what Israel chose as corrupt to the core in its very nature. Even our so called Judeo-Christian ethic of government, government of men is from the get go an irredeemable thing that has only ever seen an evil mixture of sinful man being overtly sinful and sinful man pretending to honor God.
Now, I admit I am baffled by the idea that "men of integrity and principle is the means that God will use to change the unjust laws of the land." Oh, it sounds true. It seems tru-ish. But, God moves thhe hearts of kings as He pleases and does not need "men of integrity". In fact, I would refer you to the Scriptures that tell us there is none good, no not one. None seek God. Every man goes after his own way. Nothing good dwells in our flesh. Governments of men are purely carnal. Also, please show us in Scripture where this proposition is true. If not, is is a new means of God's activity in the world and is something about Himself we do not get from His Word.
Our being salt and light never gets used as much as it does in this discussion. A Google search of the phrase with the word America demonstrates that in a measure. Voting is the American sacrament. It is the means American Christians choose of their own accord to express the light of the gospel, weakly stabbing a pencil mark onto a card listing fallen men as the priests of the salt and of the light; men who, in turn, weakly rely on arms of flesh to mark those cards to elect them so that they can go as such priests. Phineases and Hophnis all.