Most U.S. 'Christians' don't care about Middle East Christians being persecuted. Mainline Evangelicals want: Israel worship, war with Iran, support the troops, and kill the gays -- in that order.
I personally think the truth is somewhere in between what you say and what James Madison says (BTW: James Madison is a Christian too, so he's not trying to dump on Christians, although I know I've been accused of the same before despite being a Christian.) Frankly, I think your average Evangelical does care that people in the Middle East are persecuted for Christ, but don't understand that this happens many times because of, and not in spite of, military action over there. Ultimately, informing them is the only way to figure out where an Evangelical really stands is to present them with proof that these things are happening, and then still ask them where they stand. If they're radical Israel supporters, explain to them that Judaism is just as much a false religion as Islam, and that excessively supporting Israel is ultimately supporting one false religion over another, and then see where they stand. Explain to them what's actually going on in Palestine, etc.
For the average person on the street, I think more often than not its stubbornness, ignorance, or both when they oppose libertarian foreign policy. The big name Zionist preachers are another matter.