First thing to note is that the United States hardly found itself in a unique position with the famous "stubbornness" of the Moros; their holdings in Mindanao had more or less successfully resisted significant Spanish dominance until a series of naval skirmishes in the 1840s. In contrast to the petty and divided tribes of Luzon and the Visayas, the Moros of Mindanao possessed generally more standardized forms of Islamic governance and religion, and were relatively more hostile to attempts at Spanish conversion.

Second thing to note is that the date of 1902 as the end to the war was actually quite arbitrary; it was only the date when the Brigandage Act was passed, which classified any and all further resistance to American rule as banditry not subject to the laws of war. There was actually still quite a bit of indigenous resistance in the rest of the Philippines that persisted even until 1908, like the Katipunero revival of Macario Sakay and the cult of Papa Isio.

It's important to remember that while the Americans arrived in the Moro-held port of Jolo in Mindanao in 1899, widespread hostilities did not break out until 1902-1903. This was due in part to several reasons:

a.) poor translation and cultural differences which meant that the Moros and the Americans had very different conceptions of what "American sovereignty" implied, as guaranteed by a "Treaty of Paris" that they had never heard of. Some datus (chiefs) saw the Americans as benevolent and materially beneficial to their communities, while others saw them as a mere prelude to forced conversion from Islam. Nevertheless, the Americans were content to maintain the scant alliances they currently had, partially to focus on the war in the north, and partially out of naivete that the sultans allied with them represented the entirety of the Moro people. Which leads me to my second point:

b.) the "Moros" were a catch-all term for the plethora of tribal and familial factions that merely happened to be residents of a common American holding. When an American made an alliance with one datu (chief), it did not by any means guarantee the loyalty of his geographic neighbor (quite often it was the opposite). A mass military campaign to subjugate "Moroland" did not occur until the assault on Fort Pandapatan in March 1902, and even then the process of "benevolent assimilation" was to occur in fits: rebellions against American rule were made on behalf of individual datus and sultans, and were dealt with accordingly.

But as the campaigns wore on, American military officers sent to govern Moroland were acutely aware of the stark differences between the Moros and the rest of the Christian Philippines. Most of them believed that it would take two or three more generations to properly integrate the Moros into the Philippine nation - if they could even be integrated in the first place. In contrast to the westernized elites of Luzon and the Visayas, American officers were constantly exposed to the alien and "debased" practices of a people who had effectively developed a parallel state in the centuries that Spain had unified the rest of the colony. American officers soon found that the warlike and stubborn nature of Moro culture, which included among others slavery, violent blood vendettas between families, and the mandatory arming of all men with swords, was ultimately a force to accommodate, not subjugate.

Individual Moro rulers also had very different conceptions of American rule. Some early collaborators like Datu Piang were quick to ally with a coming power that they recognized was more powerful than Spain, while others resisted, like Datu Amil who was only killed by the Americans in 1913. But while this made the process of subjugation rather confusing for the Americans, it also handicapped the Moros who were never quite able to achieve any sort of unified resistance against Christian America. This was further complicated by the fact that the Americans were prepared to implement more secular and representative institutions, and that individual Moro rulers themselves had wildly varying connections and alliances to a conception of a greater Islamic cultural identity.

Thus, in addition to the threat of overwhelming force by a modern military, American officers appealed to Moro leadership by extolling the American virtues of representation and self-governance, and that ultimately their goal was to "defend" the Moros against a bigger threat: the Filipino elites in Manila that would aim to conquer them without the "moderating" influence of American rule. The Moros warmed up to this premise, and were especially interested with securing American help in establishing a "Moro province" that would not have to answer to any central authority. The Americans also built infrastructure, established markets for economic development and adjusted the abstract legal system to a form that the intensely personalistic Moros would understand. At the same time, the violent suppression of other aspects of Moro culture, like slavery and the arming of swords for all menfolk, was not entirely out of the question either. Moro leaders soon began losing hope as their negotiations shifted from American military officers to civilian government in 1913, and ultimately to the Filipino Christians in 1935 with the proclamation of the Constitution.

Sources:

- The Moro War: How America Battled an Insurgency in the Philippine Jungle, 1902-1913 by James Arnold

- State and Society in the Philippines by Abinales and Amoroso

- The Good Imperialists? by Patricio Abinales, in The US and the War on Terror in the Philippines ed. by P. Abinales and N. Quimpo