I've been able to confirm that Joseph Smith did, indeed receive a prophecy to run for the highest political office. He hoped to counteract the victimization of his people in Missouri, where they were trying to establish a community free from persecution. The Mormon vote was a strong bloc of votes and could sway many representatives of the state. But, by 1845, Joseph Smith was dead and the Saints were beginning plans to find Zion, in a place that would be their land, their government, not contingent on the United States governing bodies.
In the history of the Mormon Church persecution sent them further and further from the defined boundaries of the United States during the mid-1800s. It was hoped that what is now called Utah would be Zion, a Mormon country, but too many factors and forces made it impossible for the Mormons to hold the land as separate.
Since the failure of the Mormon hierarchy to create their own nation, the prophecy has been construed as a signal to make the Church of Jesus Christ of Latter Day Saints the religion of the United States. Thus turning our nation into a theocracy rather than a democracy.
While there are many questions regarding the initial prophecy it has not become part of the doctrine of the Mormon Church. However, Mormon beliefs aren't based in doctrinal teachings so much as in prophecy and testimony.
Remember, The Church of Jesus Christ of Later Day Saints is not even 200 years old. It is born out of something wholly American, based in community and leadership and continues to be fluid in the formation of it's belief system. Catholicism has 1800 years on the Mormons, and Judaism - 6000, which means there has been plenty of time for each to formulate doctrine. Delving into the history of any world religion you will see instances of the doctrines, prophecies, and testimonies sharing center stage, vying for the beliefs of the people.
My sense is that the Mormon Church continues to be a work-in-progress, White Horse Prophecy or not.