Perhaps I just don't know enough about Islamic and Arabic civilization, but I can't identify that period when Islam was truly united, except under the early "rightly guided" caliphs.
The rest of their long history has been filled with Fatimids, Buyids, Ummayads, Abbasids, Mamelukes, and whatever you like flavor of powers and potentates. They were only dangerous at Tours and after Constantinople fell -- and that was the Turks, not the Arabs; the Turks would have been a threat whether they'd converted to Islam or not, like their distant cousins the Mongols, who came as far as Liegnitz in the north (1241) and Ain Jalut (Well of Goliath, 1260) in the south.
Moslems were not removed from Spain until 1492. Moslems attacked Rome and controlled Sicily a couple of centuries before. Venice was in an almost constant state of war with its advocates for hundreds of years. They were only beaten back from the gates of Vienna in the 17th century and controlled Greece until the 19th (remember Byron). It is true that there was constant conflict for the levers of power which you reference but that was only over who would have the honor of destroying the enemies of Islam.
There was never complete unity nor a time without squabbling between various claimants but even so Islam has been a terrible danger for 1200 years. It rose to power only through the destruction of the Christian and Jewish cities and communities which were throughout the Middle East. Since it is more of a political movement (a totalitarian one at that) than a religion it will always be an enemy of Western Civilization.