This may have been the de facto result. However, to gather the Arabs to the call of war, it was the dream of loot, battle and general pacts of loyalty that brought the conquest. The reason this occurred, is the Arab elite sent these same warriors upon new enemies, thus removing the same strife other nations would have. I stay slightly away from the idea that the Umayyads replaced former empires until the Abbasid Period in which this transformation was complete.
Well, in hindsight, we may regard the Umayyad period as one of transition. But of course, they did not know that. Clearly, the Umayyad Empire was a conquest state, where loot and its management was an essential part of what kept the elites (sort of) together. It is worth noting that the early Umma experienced three civil wars, a the onset, midst and end of the Umayyad rule respectively, so, they had a fair amount of strife (again, not unlike the Mongols in this).
Also, it may be useful to separate the Sufyanid and Marwanid phases of the Umayyad empire: under the Marwanids, and especially after 'Abd al-Malik and al-Walid, the Umayyad state was a lot more structured and self-conscious, and it seems clear from Imperial inconography that a perception of inheritance of older empires existed (sure, we have more or less nothing about Sufyanid Imperial iconography and architecture, the big stuff having been built only since 'Abd al-Malik as far as we know).
I fully agree that loot, warrior ethics and loyalty were the basic elements of the conquest, underpinned by the new (and yet not fully codified) faith. It is also important to note that faith was likely, as you point out, not the key element in itself (not before 'Umar II reign at least) as we know that a significant portion of the conquering armies was composed of Arab Christians and even Persian mazdean forces, plus other significant contribution that were neither Arab or Muslim.