Just because there are Mormons who vote Democrat doesn't make it right. It really boils down to two things for Mormon Democrats. Either:
1)They don't know the doctrine.
2)They don't believe the doctrine.
In my experience, it's usually the latter.
So, Brother Nibley didn't know the doctrine, nor did Hugh B. Brown (even a more liberal democrat than Nibley was), nor Pres. Faust--because these men certainly believed it. Today there is Elders Marlin Jensen and Steve Snow of the Seventy. Elder Jensen was church historian, and he has been replaced both Elder Snow--and both were well known democrats, both very active in the Utah Democratic party before their calls. These men certainly believe the doctrine...so it must be they don't understand it? I think most Utah LDS would say that Harry Reid, Democrat from Nevada and Senate Majority Leader, doesn't understand the doctrine nor does he really believe it; but I would disagree. When he said at the BYU Law School Commencement a few years back that "I am a Democrat, not in spite of my LDS beliefs but because of them," I believed him. It is the way I feel. But I am not bothered in the least if one of my Republican brethren in my High Priests Group feels differently.
There is not one way and one way only to read and understand "the doctrine", i.e., the scriptures, especially when it comes to politics. Hugh B. Brown and Ezra Taft Benson are good examples of this. Both were apostles. Pres. Brown was in the First Prez, later ETB would become, for a very short time, president. I think it is safe to say both understood the doctrine, but they drew different conclusions when it came to politics. ETB was not just right-wing Republican, he openly advocated the John Birch Society, and his son Reed was later chapter president for Utah (I think). He openly opposed Civil Rights, said it was a communist tool, etc., etc. Hugh B. Brown, on the other hand, a known Democrat, not only strongly favored Civil Rights, he lobbied every church president he served under to extend the priesthood to the blacks. Etc, etc.
I don't think you can say that one of these men was right and the other wrong, or one understood the doctrine and one didn't. It has so much to do with background, one's experience. Harry Reid grew up in Searchlight, NV. His mother took in washing for the local brothels, and his father was a drunk and ne'er do well. I think he later committed suicide. Reid learned to fight as youth, both literally and figuratively, and it is no wonder that today he has a lot of sympathy for the underdog. Hugh B. Brown was much the same way, if I am to believe certain statements he made. Omar Kader, former BYU faculty and active Mormon today, is a Palestinian and a Democrat. Given his background, it would be hard to think of him as a Republican, or espousing Republican ideals.
And that's okay--at least I think it is. "There is too much sameness among this people," Brigham Young said. "Let there be no stereotyped Mormons."