This is so full of misinformation it's hard to know where to begin. There is one kernel of truth here - that the far, or al-aqsa mosque mentioned in Sura 17, is never identified as being in Jerusalem; in fact, most Muslims also would have placed it in Medina until around the 10th century CE. However, the Muslim view of Jerusalem as a sacred space is hardly tied explicitly to the idea of it having been one of the sites of the Night Journey. That's certainly an important factor now, but it's not the only one, and it would have never entered the mind of abd al-Malik when he commissioned the Dome of the Rock.
Umar didn't take the city with his "rampaging hordes", except in the loosest conceivable sense: Patriarch Sophronius, the Christian leader of the city under the Byzantines, surrendered it to him, without a drop of blood being shed, in return for Umar's promise that Christians would be able to practice their religion freely. The Church of St. Mary of Justinian didn't exist on the Temple Mount. The Byzantines had turned the entire mount into a garbage dump for the city. The Al Aqsa Mosque, on the southern edge of the Temple Mount, was built around Umar's time, but as a temporary structure, out of wood, and it certainly wasn't called that yet.
The reasons for the construction of the Dome of the Rock are unrecorded. The central reason, in all likelihood, was the Muslim belief, acquired from the Jews, that the rock on the Temple Mount was literally the center of the universe; the place where God had stood on the moment of creation, and created the world around him. Incidentally, this can also be seen in medieval Christian world maps that show Jerusalem as being at the very center of the universe.
As an aside, basing these assertions on the idea of what Mohammed was thinking when he wrote the 17th sura is unlikely to convince many Muslims of anything. One of the basic tenets of Islam is that the Qur'an is the literal and inerrant word of God, as dictated to Mohammed by the angel Gabriel.
Muslims and Jews fighting over Jerusalem is a very, very recent phenomenon, one that essentially started in the 20th century. For most of the history of Jerusalem under Muslim rule, the two faiths existed in relative harmony, in the face of hostility to both from Christianity. For instance, when Umar first took the city, he re-admitted the Jews to the city; the first time they had been allowed in, except for once a year to mourn at the Western Wall, since Jerusalem came under Byzantine domination. When Christians took Jerusalem in the First Crusade, one of their first actions was to expel the Jews, and when Saladin retook it, one of his first was to re-admit them.
The point, anyway, is that the picture is much, much more complex than any piece of propaganda can present. The history of Jerusalem is one of thousands of years of intertwining faiths and beliefs, all of them evolving over time, and none of them anticipating the conflicts and wars of the present.