If you are interested in the subject I would strongly recommend a book (which just came out this year) titled "Unworthy Republic: the Dispossession of Native Americans and the Road to Indian Territory" by Claudio Saunt. I will be basing my answer on the evidence provided in this book.

I cannot speak as much to "Wounded Knee" and the situation that came much later in the nineteenth century which had a variety of different causes relating to the idea of Manifest Destiny. I will instead focus on the separate set of events that played out during American expansion into the region of the "new South" in the early 19th century.

To answer your first question: U.S. military intervention to drive out the Chickasaws, Choctaws, Creeks, Cherokees, and Seminoles and relocate them west of the Mississippi River was an initiative that was primarily driven by the slaveholding aristocratic class of the old South. This planter class desired the fertile land on which Natives resided, and needed to remove them to expand cotton production in order to meet demand and increase their profits. Prior to the election of Andrew Jackson to the presidency, prominent Southern aristocrats and politicians would work to dehumanize Natives and proclaim "paternalistic" and "humanitarian" reasons for relocating them. They peddled myths about Native Americans living in "desperate conditions" and argued that resettling them toward more "fertile" lands in modern-day Oklahoma would somehow solve all of their problems (Saunt 2020, 14). This wasn't true, though numbers of Native peoples had been dwindling for a variety of unrelated reasons (disease, etc..) ever since Europeans first arrived on the continent. Others within the Southern planting class simply made racist appeals and argued that Natives and whites could never coexist. All of this served to justify forced removal in the eyes of the Southern populace- and for the most part, it worked.

When Andrew Jackson finally took office, these Southern planters found a sympathetic administration that was willing to send federal troops to assist in the removal process. This inevitably resulted in several short-lived wars to drive out these indigenous tribes- specifically the Black Hawk War, the Creek War, and the Second Seminole War (which actually dragged out quite a bit because federal troops struggled to carry out military operations in the swampy terrain- as a result,a small community of Seminoles remains on the Florida peninsula to this day).

However, to get at your second question, yes there was a significant amount of public debate surrounding the federal government's decision to back these operations militarily. The debate was largely spurred by Native American activists who appealed to Northern citizens, similar to the way that Black abolitionists would later appeal to Northerners to join the anti-slavery cause. In 1829, Jeremiah Evarts, then the corresponding secretary of the American Board, published twenty-four essays criticizing the Jackson administration for violating U.S. treaties with Native Americans and initiating a policy of removal. Evarts became a major thorn in the administration's side- his publications found a wide audience and it spurred a flood of petitions to Congress from states all across the North. Of these petitions "signatures ranged in number from two to three to several hundred" (Saunt 2020, 65). Due to their heavy influence in government, the Southern elites won out over the marginal religious and civil society groups in the North who had sided with Native activists. Thus, they were able to expand their cotton kingdom toward fertile new grounds while enriching New York City cotton investors in the process.

Some Cherokee leaders were even able to bring the issue of sovereignty all the way to the Supreme Court. The case of Cherokee Nation v. Georgia (1828) ruled that the Cherokee Nation was a "domestic dependent nation" and the test case of Worcester v. Georgia (1832) actually ruled in favor of the Cherokees- and declared that non-Native white men could not encroach upon native lands. In response, Jackson famously ignored the court's ruling and went forward with Cherokee Removal in 1838, the last effort within the series of removals which are encapsulated under "the Trail of Tears".

I hope this helps! (and hopefully, someone else can respond to your question regarding Wounded Knee)