I do think that the Republicans have attempted to co-opt conservative Christians. I am dismayed by the extent to which they have been successful.
There are two separate but vital threads in Christian moral thought, but very few churches or individuals manage to weave both of the threads together very well.
As Dave K has repeatedly reminded us, there is a strong tradition of advocacy for, support of and identification with the economically deprived. Those who focus (often exclusively) on this thread tend to land on the political left, at least on issues of economic justice. Many of these people do not really believe that God is still active in the world in any meaningful way, and many of them are as thoroughly liberal theologically as they are politically. Many of these people have all but rejected the Scriptural standards for personal morality and focus instead on societal justice.
As others have repeatedly reminded us, there is a strong tradition of evangelism based on the premise that all are personally subject to God's moral standards, all have sinned, all are condemned and all stand in need of redemption offered via the person and work of Jesus Christ. Those who focus (often exclusively) on this thread tend to land on the political right, at least on issues of personal responsibility and personal morality. Many of these people do not acknowledge the very real social injustices in our world, and even if they do acknowledge them they may believe that social justice has nothing to do with the Gospel. Some of them are so tightly focused on issues of personal morality that they easily lose sight of the other demands of the Gospel.
IMO, it is not possible to be a truly orthodox Christian without maintaining an emphasis on both aspects of Christian moral thought. Holiness matters, salvation matters, and social justice matters. Furthermore, orthodox Christians must be very careful to make sure that they do not become identified too closely with any secular institution and particularly not with political entities such as our leading national political parties. Political power often requires a degree of moral 'flexibility' that is, in many ways, not compatible with true faithfulness to the Gospel. Christians who attempt to enter too deeply into the secular political arena do so at their peril.
I am not saying that all politicians who claim to be devout Christians are misrepresenting their faith. I will say that I suspect most of them are (either knowingly or unknowingly) less faithful than they claim to be. God has, undoubtedly, called some of His servants into the political realm, but I suspect that most of the nominal Christians who enter the political realm and play by secular rules will quickly lose whatever grounding they had in real faith.
As to why mainline churches are not raising more hue and cry about this? I'm not sure I can express my opinion without offending a large number of people, but my suspicion is that most mainline churches are not raising a hue and cry because they lack the passion required to make any meaningful statement at all. My opinion is that many mainline churches have become little more than social clubs filled by people who attend out of habit but who lack firmly rooted convictions about much of anything. Absent deeply held convictions there is little chance that these churches will influence their own members, much less society at large. I would gladly be wrong about this.