The story of the American buffalo's depletion is a bit more complex than usually assumed. The US Army did land the finishing blow, but the population had already been severely crippled through other factors.
The tragedy of the booming industry of livestock trading among the Indian tribes was its degree of ecological decimation. The growing domesticated herds of horses reduced the frequency of grass feed for bison. Comanches worsened the situation by opening their hunting territories to foreign parties for trading privileges, and the Cheyenne and Arapahoes began hunting bison more aggressively to increase trade with Americans. Bison herds dwindled.
--- “As their subsistence economy crumbled with the dwindling bison herds, Comanches lost the ability to restore their numbers after disease epidemics. Their population fell from some 20,000 in the 1820s to fewer than 5,000 in the 1860s, forcing them to scale down their political and economic ambitions. They dismantled their trade empire, yielded large tracts to the Anglo-Texan ranching frontier, and, together with Kiowas, Cheyennes, and Arapahoes, concentrated on opportunistic raiding along the Santa Fe Trail. The decrease in the bison herds also fueled internal rivalry, splintering the Cheyennes and Arapahoes into bitter factions that struggled for the shrinking resources and for political power. Starving and desperate, the southern Plains tribes began to raise even more horses, which they increasingly used for food. With up to eighty horses per family in the 1850s and 1860s, they were in the process of becoming bona fide pastoralists who relied on domesticated animals for much of their food supply” (Hämäläinen, par 23).
Now some background on the US government's relationship with the Plains tribes. In 1851 a treaty was signed among the tribes with the US government that would allegedly establish "an effective and lasting peace." The treaty affirmed that said land was Indian territory and that the US would work to protect the signing tribes from aggressive white settlers or even other hostile tribes. In exchange, the US would be allowed to build forts and safely have travelers sojourn west without threat of an attack. Unfortunately, the discovery of gold in Indian territory led to an influx of migrants that were violating the territorial claims. The Treaty of Fort Wise in 1861 then limited their territory to less than a tenth of the claim in the original treaty. Tribes began to rebel, people were killed, and a series of conflicts between the US and subjugated Indians ensued.
The US Army was wary of the tribes and a sort of fearful panic overtook some commanders. It's my understanding that there was no official decree to start killing bison to starve the Indian tribes, but rather it was the decision of individual commanders and soldiers who sought to deal with the "Indian problem" themselves. It seemed more effective to limit their numbers, not through direct warfare but rather the elimination of the primary food source.
After the Civil War the US Army expanded its influence to the southern and central plains. They knew the key weakness of Indian survival was its reliance on bison herds. The US Army hired professional bison hunters and systemically slaughtered captured Indian horses. American hunters would kill off the last of the bison from the central Plains by 1868.
The army then killed off the bison herds of the Comanche and Kiowa territories by the fall of 1874. The following Red River War exterminated people who had lost their ability to feed and defend themselves.
The death of the American bison wasn't solely fueled by "anti-Native hatred." Like Apaches killing off the Jumanohs, the Comanches subjugating the Apaches, or the Flathead invading Blackfeet territory, the actions were motivated by acquiring new territory, the possibility of resource-rich lands, and the ever-present fear that the "other" is a threat to their existence or prosperity. Of course, there were racist sentiments held by people towards the Indians, and no doubt these emotions helped fuel the public and the society at large to lend support towards the elimination of particular tribes. But the history of the Native-European conflict is long and complex. When one hears of an attack on a wayfaring trade envoy or a small group of European migrants, it's easy for the public to fan the flames of anti-Native sentiment, and no doubt these emotional responses helped to spur the mission of resource expansion into the west and south.
Hämäläinen, Pekka (2003). "The Rise and Fall of Plains Indian Horse Culture". Journal of American History. 90 (3): 833–862. doi:10.2307/3660878. JSTOR 3660878. Archived from the original on 28 February 2009. Retrieved 8 September 2020https://web.archive.org/web/20090228175333/http://www.historycooperative.org/journals/jah/90.3/hamalainen.html