It is difficult to answer questions about popular memory or perceptions of history, but I can give it a go.

The Trail of Tears was one of many Indian removals from east of the Mississippi during the early 19th century. Following the Revolution, Americans flooded over the Appalachian Mountains (which Britain had tried to prevent during the colonial period, with mixed success). The population was growing rapidly, and there was an ever-growing hunger for land. This manifested in a broad movement to evict all Native Americans from east of the Mississippi. The Cherokee were one of many groups removed in this time. While we commonly speak of it as a singular event, it was actually dozens of these removals as more and more people, Cherokee and otherwise, were forced Westward.

Jackson made indian fighting and indian removal a part of his political ideology. He fought in the Creek and Seminole wars in the decades before his presidency, as well as the war of 1812, in which many Indians rallied with the British to try to stop western American expansion. Jackson orchestrated a massive transfer of Creek lands to the US as part of the treaty that ended the Creek War, for example. This has made him the figure most associated with removal, which was a popular policy and a component of his populist platform.

The Trail of Tears is the most prominent of these removals, so it has become representative of a larger contemporaneous movement of removal that spanned from the Gulf of Mexico to the Great Lakes.

There are, as you indicate, many parallels between the Trail of Tears and the Navajo Long Walk. Both were brutal overland marches, manifested from genocidal federal policies towards Indian people. I would posit that the greater prominence of the Trail of Tears in the American consciousness is largely due to four factors: the first two are what I already covered, that Jackson was a huge proponent of indian genocide and removal, and that the Trail of Tears was simultaneous to other removals.

Another is that the Cherokee in particular were highly integrated into the young American nation, meaning the political-spacial reality, not just the land. Removal of the Cherokee was highly visible (and popular) in the South. Thus gave it a place in popular memory.

A final factor may be Lincoln's role in the civil war and assassination has simply elevated his esteem so highly in the American consciousness that it is less popular and interesting to many Americans. To hold him accountable for the Long Walk goes against his (literal) enshrinement into the pantheon of Great Americans, where Jackson has long been a contentious figure in American history and the target of various criticisms.

Sources: Colin Calloway, New Worlds For All: Indians, Europeans, and the Remaking of Early America (2013)

Claudio Saunt, Unworthy Republic: The Dispossession of Native Americans and the Road to Indian Territory (2020)

I'll use a well known and exemplary case to elaborate on on why this is:

The Cherokee who were removed under Jackson were a relatively politically united group who had adopted many European customs, including large-scale sedentary agriculture and somewhat of a European style governance structure with an elected Principal Chief. Some of their big planters even bought black slaves and had white indentured servants. Their territory was highly reduced through a series of treaties, but was nonetheless still very substantial and valuable, particularly that of the group of Cherokee in the north of Georgia, as they not only had lots of great arable land but also later found that they had proven gold deposits. This situation had emerged because after fighting with the British in the War of Independence and losing, they agreed a peace with the US that ceded most of their land in exchange for the US guaranteeing their independence and essentially allowing them to coexist with the US as a quasi-independent nation while surrounded by US-annexed territory. Ostensibly, it was the duty of the US federal government to protect them from violations of their independence & land rights by private individuals and state governments.

For all intents and purposes, they'd become a very friendly neighbouring country (at least friendly to the US federal government), and were the exemplar of the supposed 'success' of Jefferson/Washington's 'assimilation' policy, whereby Indians would be allowed to live 'peacefully' alongside white Americans provided that they agreed to give up most of their land and take up agriculture/other European ways on much reduced areas.

Jackson however was a long-time anti-Indian advocate who believed them to be innately inferior and incapable of doing what the Cherokee had basically already proven they could do: live 'like Europeans'. So under his presidency he implemented a federal government policy of forced removal, doing basically everything possible to remove Indians as far West as possible, regardless of whether they'd been good friends of the US government or not.

Settlers had long been encroaching on Cherokee land in northern Georgia, and when they went to Jackson's federal government for help, they were ignored, which was a tacit endorsement of the invasion onto their land that the Georgia state government+settlers were engaging in. Without federal government help, they had no options, as if they resorted to violence to defend their land it would have been used as a justification to exterminate them physically. They were overall highly outnumbered and had little chance of winning an actual war against combined state militia and federal military forces. For example, in 1836, the Creek attempted to resist settler encroachment and federal government inaction to enforce their treaty land rights with violence, which resulted in an overwhelmingly disproportionate response, including the intervention of federal troops, massacres of hundreds of their people, and forced deportation for all survivors anyway. So that was the fate that probably would've awaited the Cherokee too if they'd (justifiably, I might add, since legally and morally they were clearly in the right) defended their land.

Eventually this culminated in a fraudulent 'treaty', signed by a tiny group of Cherokee with no power to represent their nation, where they 'agreed' to give up their land and be deported further West. This was so ridiculous that it was basically the equivalent of Oswald Mosley and an entourage travelling to Germany and signing a treaty that ceded all of Britain to the Nazis.

Nonetheless, the federal government then knowingly used their fraudulent treaty as a justification for the forced removal of 10,000+ Cherokee, practically none of whom actually wanted to leave, resulting in the Cherokee Trail of Tears and a long-term shift from the (already genocidal and dishonest) assimilation policy to one of forced removal. Which Jackson also duly carried out on many more groups, including the other four of the fairly assimilated 'Five Civilized Tribes', who all have their own 'Trail of Tears' as a result. All of this was incredibly 'illegal' even under the US' own federal law, since all of these nations had clear treaties with the USA which were being violated. That didn't matter because when it came down to it, the government always ended up siding with settlers over Indians, even when the Indians were proven allies.

This is why Jackson is so singled out: Lincoln and those before him who also carried out removals were basically doing Jacksonianism, simply continuing the pre-existing federal government policy that Jackson had pioneered. This led to the reservation system in the second half of the 19th century, as Western deportation was no longer an option. After the annexation of the West coast, there was no Western land left to deport them to and a longer term 'solution' was needed.

So 70+ years of genocidal 'Indian policy', including Lincoln's, could be considered a result of Jackson's actions. In reality it can't be reduced simply to him, as the US had always been a project of continental expansion that necessitated the disposession and genocide of Native Americans, and Jackson's hatred of Indians and views on what ought to be done to them were completely normal among Americans who lived in the West. But he was the first president who dropped all the previous auspices of 'concern' for Indian welfare, took the mask off, and just admitted it, setting the norm for those who came after him.

It may have been possible that a USA without Jackson or someone like him could have allowed some independent Indian nations to remain, though I think that would be highly unlikely given the land hunger and Indian hatred that was so common among those on the frontier. For that to happen, the federal government would've needed to have been willing to use military force against white Americans to protect Indians - something that even sympathetic Eastern elites were unwilling to do. The government of and for white people always chose white people in the end.

A few sources that cover these events & Jackson's shift from earlier policy:

The Cherokee Nation and the Trail of Tears, Theda Perdue & Michael D. Green.

The Cherokee Removal: A Brief History with Documents, Theda Perdue & Michael D. Green.

The Politics of Indian Removal: Creek Government and Society in Crisis, Michael D. Green.

Abuse of Power: Andrew Jackson and the Indian Removal Act of 1830, Afred A. Cave.