The problem is the perfect, ideal view of Islam you have and project is like the perfect, ideal view of every theism or human system for that matter, a flight of fancy, a pipe dream that doesn't, nor ever will exist in the real world. Behind the walls of churches, mosques, temples, monasteries, buildings, court yards, homes, etc it can occasionally be made to work but out in the real world where politics, economics and power come into play, individuals, groups and governments make it what it is, not desired ideal theory.
But it did.
Rashidun Caliphate - Wikipedia, the free encyclopedia
Yes, it worked off of the conquered's resources. The below is from the site you gave. It looks like the military was working full time killing and collecting others' wealth. It doesn't look like this system of governence can stand on its own, it must take/steal from others to survive.
Abu Bakr desired Umar to be his successor and he persuaded the most powerful of the followers of Muhammad to go along. Umar was gifted both militarily and politically.
Umar continued the war of conquests begun by Abu Bakr. He pressed into the Sassanid Persian Empire itself, but he also headed north into Syria and Byzantine territory and west into Egypt. These were some of the richest regions in the world guarded by powerful states, but a lengthy war between the Byzantines and Sassanids had left both states militarily exhausted. Islamic forces easily prevailed in war against the two states. By 640, Islamic military campaigns had brought all of Mesopotamia, Syria and Palestine under the control of
Rashidun Caliphate. Egypt was conquered by 642 and the entire Persian Empire by 643.
Umar, however, was one of the great political geniuses of history. While the empire was expanding at a mind-numbing rate beneath his leadership, he also began to build the foundations for a political structure that would hold it together. Umar did not require that non-Muslim populations convert to Islam nor did he try to centralize government, as the Persians had done. Instead, he allowed subject populations to retain their religion, language, customs, and government relatively untouched. The only intrusion would be a governor (amir) and a financial officer called an amil.
(I believe we would call that a "tax collector")
Umar's most far-reaching innovations were in the area of building a financial structure to the empire. He understood that the most important aspect of the empire was a stable financial structure for the government. To this end, he built an efficient system of taxation and brought the military directly under the financial control of the state. He also founded the Diwan, a unique Islamic institution. The diwan consisted of individuals that were important to the Islamic faith and the Islamic world, such as the companions of Muhammad. Their contribution to the faith was so great that they were given pensions on which to live,
(supported by others' resources) which freed them up to pursue religious and ethical studies, and thus provide spiritual leadership to the rest of the Islamic world.
Umar established many Islamic traditions, including the process of collating the Quran. Among his most lasting traditions was the establishment of the Muslim calendar. Like the Arabian calendar, it remained a lunar calendar, but Umar set the beginning of the calendar to the year in which Muhammad emigrated to Medina. This, as far as Umar was concerned, was the turning point in Islamic history.
Umar was mortally wounded in an assassination attempt by the Persian slave Abu Lulu Fieroz,
(a disgruntled employee?) during morning prayers in 644. Before he died, Umar appointed a committee of six men to decide on the next caliph—they were charged to choose one of their own number.
Looks like this "perfect system" was busy conquering neighbors and collecting war booty.
It appears islam cannot survive without "conquering" (spending other peoples' resources).