Partick, I think you accurately point out that current day Americans, including but not limited to African Americans, Asians, Hispanics and other "races," (all of whom had ancestors involved in actions taken against Native Americans in that era) cannot be expected to allow a suicidal invasion to occur now because it happened to the prior residents of this geographic area (the current US). It is worth adding that we do in fact allow 10s of thousands of immigrants into the US per year.
The particular issue with the Syrians is of course a matter of trust - these folks come from a region with well-known animosity toward our country and not without substantial justification. Thus the question is do we allow people in who may want to destroy us. I think many prudent minds are having second thoughts about this while the collectivists, including our collectivist-in-chief, are not (or perhaps not having any thoughts beyond the collective).
If the Native Americans could live that era over again, if they had any idea what would happen to them, would they do even more to keep the "immigrants" out. Hell yes, they would. I think there are some strong parallels between their position then and ours now.