I think first we need to consider what we mean by sympathy in this context. Let us not mince words; What happened to the Native Americans was an atrocity. It was genocide and everyone involved should be condemned for their actions, but when we are considering questions of historical attitudes about things like sympathy it is important to realize that modern standards of what a truly sympathetic person should have done are very different from what a contemporary sympathetic person would actually have done. It seems to me a bit myopic to define sympathy so narrowly that only John Brown could be said to be the one true abolitionist when far less sympathetic people got more done.

People who are sympathetic to a different group of people are very capable of doing incredible harm to those people, which was frequently the case. By modern standards we recognize that being sympathetic to another cultural group requires being respectful of their culture as well as their physical integrity. When considering historical motivations we must be able to recognize both when someone committed an atrocity as well as how they believed they were doing something good. For a deeper exploration of this concept I recommend Eichmann in Jerusalem by Hannah Arendt in which she coins the term "the banality of evil."

Now, on to the question itself.

What sympathy there was, was not contextualized as a condemnation of American colonial policy. From the perspective of Europe and Americans on the east coast, the fate of Native Americans was often regarded as inevitable and tragic. This is known the noble savage myth. From their perspective, that was sympathy but it was really a kind of helpless passivity that only perpetuated the genocide; a concept that they did not really have the verbiage to describe at the time as the term was only popularized in 1948 by Raphael Lemkin to describe the Holocaust and Armenian genocide. An example of this attitude is present in the 1831 edition of Frankenstein by Mary Shelley in which we can see a simultaneous sense of tragedy and inevitability:

"I heard of the discovery of the American hemisphere and wept with Safie over the hapless fate of its original inhabitants."

There were many different reactions to this conception of inevitable tragedy. One of the most famous was the sentiment popularized by R. H. Pratt in the phrase: "Kill the Indian in him, and save the man." This is the attitude that led to Indian boarding schools in the US as well as Australia and Canada. It was believed that the way to save Native Americans was to educate them to operate in white society; in other words, force them to assimilate. For some people, this notion of civilizing indigenous people was born of a profound sympathy that only extended to the life and limb of a human being with no other considerations. It compromised on physical genocide with a cultural one. In the linked speech, Pratt speaks at length about how his belief in assimilation came from close personal encounters with Native Americans. While he is extremely condescending towards native people, he is deeply sympathetic to them in the context of the time in which he lived but the system that he ended up pioneering was perhaps one of the most destructive and traumatizing parts of the genocide.

Were there individuals that outright took the side of Native Americans? There were certainly some but it is never so simple as picking a side. There are some accounts of people, typically women, who were captured and integrated into the tribal group. I won't go in to too much detail on each of these because I am running out of time to write this. Mary Jemison is a good example from the 1700s. Olive Oatman is a later example from the early 1800s. I remain somewhat skeptical of their accounts because both monetized the experience by publishing books after returning to white society. Cynthia Ann Parker's story seems more grounded to me because she was clearly fully integrated into the culture of the Commanche tribe by marrying and having children, and fell into deep depression after she was forcibly recaptured and returned to her American family. That said, I am not familiar with the historiography around their accounts so if someone else could comment on that, I would be much obliged.

Like the top comment says, I think that "sympathize" is a very elastic word here. In Canada, initial European explorations of the interior were driven to a significant degree by the search for new fur trading partners as well as more generic exploration.

It was not unusual for fur traders to marry indigenous women and form kinship ties with their bands. Some even abandoned their lives in eastern Canada and settled permanently in the west, with a foot in both cultures. Now, there's no question that an element of these marriages was commercial since they were part of a formalization of trading relationships. But at the same time, these people married, had children, and sometimes even brought their indigenous wives back to "civilization" in the east. Lots on the subject in Sylvia Van Kirk's 'Many Tender Ties' though the book begins to show its age.

David Thompson is an interesting example. He was a clerk and a cartographer who mapped millions of square km using hand instruments while traveling by canoe in the North American interior. He kept extensive notebooks which he later published as a series of memoirs. He looked at indigenous spirituality 'sympathetically' in the sense that he tried to identify their cosmology with the Christian one, rather than condemning them as heathens. He also deliberately destroyed liquor barrels he was asked by his employer to take into the Rockies because he feared its impact on indigenous cultures. Both of these are complicated gestures shot through with paternalism, but they are certainly sympathetic.

He also married a Metis woman, Charlotte Small, whose mother was Cree. It's a bit creepy since he was 29 and she was 13, but they stayed married for a staggering 57 years. This is against an earlier stereotype that fur traders slept casually with indigenous women while in the west and then returned home.

What I think Thompson's example (and others) shows is (and maybe I am going to get pilloried for generalizing, let's see) that in contact zones, on terms of relative equality, Europeans were at least capable of being respectful of indigenous cultures. Where European settlement took root, indigenous people were pushed to the literal and metaphorical margins, limiting social contact between them and settlers. This is still the case today in many places in North America. Interesting, white women played a role in socially marginalizing indigenous people, especially indigenous woman, something that comes up in Van Kirk and also Adele Perry's 'Edge of Empire'. In general, white women established their place in colonial society by pushing indigenous women down.

Then again, in her suggestively titled memoir 'Roughing it in the Bush, the abolitionist and Canadian pioneer Susannah Moodie (nee Strickland), speaks much more favorably of "Indians" than she does of Americans. She mentions specifically that her husband invited "Indians" to eat at their table rather than seating them with their servants. Which is an interesting comment on class and race in Victorian Canada!

Anyway, as I said above, I think if you look in contact zones you'll sometimes (but by no means always) find a friendlier relationship between indigenous people and Europeans. Another book to check out on the subject of evolving relations in Canada is Fisher's 'Contact and Conflict' which influenced my thinking for this post.

You might do well to begin with a few different texts, which adequately answer some of the questions you have about “attitudes” during European expansion across North America. Start with Colin G. Calloway, Alan Taylor, and Claudio Saunt. Saunt’s UNWORTHY REPUBLIC, responds to your queries about attitudes during the aftermath of the 1830 Indian Removal Act. One politician during that period who tries to motivate his contemporaries to resist such native expulsion was Henry Clay from Kentucky. Privately, Clay did not care much about natives, preferring they abandon their ancestral homelands or “civilize” into white culture. But publicly, in Congress, he derided President Andrew Jackson and his administration’s callous use of the American state and colonial settlers to push Indians westward from the Southwest US (specifically, Georgia).

Here’s the catch, though. Clay only did so to undermine his political foe, Jackson, and Jackson’s Democratic Party. Once again, he, like many of his contemporaries, treated natives ambivalently. Americans generally held that natives would “inevitably” waste away. Their ways of life were so antithetical to the rapid technological civilization the US and Europe was cultivating, that they would assimilate and become white, or die. This led many opponents as well as proponents of Indian Removal policies to treat natives paternalistically (condescendingly), either trying to “save” them by introducing nations to white culture, or eradicating them through brute force.

The first few pages of Saunt’s work address many of these key themes of state-sponsored expulsion and the problems with thinking natives would “inevitably” fade away. Saunt also addresses the idea of historical contingency in his work; that is, past Americans made choices. Nothing is inevitable. And like the issue of slavery, policymakers willfully made choices they knew would harm native nations based upon their prejudices.

Seriously, read those authors’ works.

Not to correct anything here, but some supplemental points.

It's worth keeping in mind that not only is "sympathizing" an elastic term, but so is the idea of a Native "side", and to the extent that a side existed it was largely itself a construct of or reaction to colonization. Which is to say, Native communities themselves didn't necessarily see themselves as sharing an identity with or sharing common interests with other communities for long stretches of time. While this has somewhat changed with modern politics, but not uniformly: I've had conversations with White Mountain Apache tribal members who are...amused, shall we say, when white people speak sympathetically of Geronimo, whom they still have a dislike for and are proud to have participated in the capture of.

Even within tribes, nations and communities, there were and are intense political debates about what sorts of policies should be followed. The "Five Civilized Tribes" are probably a great example of this: this is a big oversimplification, but generally the white Americans sympathetic to them not being remover in the 1830s would have been less sympathetic to their legal owning of enslaved people, and vice versa, while both sides would have seen their assimilation and dilution into white American culture as a good thing (those more hostile to Indians would have questioned how well such assimilation could "take"). Even with individual activists like Zitkala-Ša, you had Native people who campaigned for greater rights while also supporting things like the Curtis Act which broke up tribal governments in favor of assimilation: so even a Native person who was at the forefront of activism in the 1900s would be downright reactionary and assimilationist to later activist generations (the Curtis Act itself is named after its sponsor, enrolled Kaw member, then-Congressman and future Vice President Charles Curtis). It's all very complex.