I don't like the term "Native American." I am a native American, as are most of the people I know. Use of the term "Native American" for American Indians implies that the rest of us don't belong here. As with most politically correct term, there is a deliterious message behind the term that I reject, and I therefore don't like the term.
Implying that Native Americans were here first doesn't mean the rest of us don't belong here. Just like saying that Whites were here before Blacks, Asians, and Hispanics doesn't mean the latter groups don't belong here.
It is simply a matter of plain correctness, not just political correctness. I'm not sure there is a "deliterious message behind the term", as you say.