Your position seems to be the position of most Christians.
Will it work if we elect Atheists, then pray that the Atheists will change their ways? By electing a person, we are endorsing their Atheism. How can anyone justify voting for a person that advocates abortion and homosexuality, or voting for anyone that is part of a political party that advocates abortion and homosexuality?
Surly it is good to pray for those in leadership and authority, but it is better to elect people that listen to God, so that God can have an effect through the Holy Spirit. I do not believe people that support abortion and homosexuality are listening to God.
Some would say they have totally satisfied there Christian obligation if they vote and pray, but by the time they get to vote, the candidates available are not even Christians, so you are voting for the lesser of two evils.
I believe the Restoration of America to being a Christian Nation means every Christian must be mindful of politics and many must devote their life to being political missionaries, by devoting their life to political life. See the thread "Christian Nation Restoration".
In the discussion of Republican verses democrats, I make the following statement.
Republicans have yet to prove to me that they totally support Christian principles, but democrats have proven they do "not" support Christian Principles.
I believe the Republican Party would be the best possibility for Restoring America to being a Christian Nation, but only if Christians first take over the Republican Party.
For most of my Christian life I debated whether a Christian had an obligation to be involved in politics, but as I have seen Atheism and Socialism (symbiotic relationship) grow in America, I decided being involved in politics was a necessity. Although challenged by the world, I have been at peace with God.
I believe the Bible is an instruction book for both individuals and nations. Neither individuals or nations will prosper/survive if the Church and State are not under the leadership of the Lord and following the commandments/doctrines of the Bible.
I believe Solomon summed up the whole thing.
Ecclesiastes 12:13-14, "Let us hear the conclusion of the whole matter: Fear God, and keep his commandments: for this is the whole duty of man. For God shall bring every work into judgement, and every secret thing, whether it be good, or whether it be evil."