Brian, In answer to your question.. NOTE: This is just MY OPINION and FEELINGS...
The term Native American was coined during the civil rights movement and was just a label placed on our American Indians to make it politically correct.. just like African American. They didn't ask the tribes what they wanted to be called - just decided this was politically correct. As such it's demeaning. It strips away the dignity and respect of the American Indian and is designed to make "us" feel better.
In addition, a Native American is anyone born inside the American borders -- including Central Americans and South Americans.. This is demeaning to our American Indians -- it's like saying they are nothing special.
These proud people have been called American Indians for hundreds of years - they fought, died and lived as American Indians. They entered into treaties - which were violated as American Indians. Regardless of the fact that Columbus named them Indians because he thought in was in India -- this is the label they have been referred to for years. You can't just erase history.
When a person refers to someone as an American Indian - you know exactly who they are referring to; a mental image of a proud man standing beside or sitting on his horse in full Indian dress. You can picture the feathers in the chiefs headband and know that he is leading braves that are willing to fight and die for their way of life, family and land. This is an American Indian.
The younger American Indians have grown up being referred to as Native Americans and have been taught they should feel ashamed of their heritage and thus prefer to use the term Native American. Now the older ancestors - they prefered the label they fought, died and won freedom as: An American Indian.
I could go on -- but this should provide you the answer. American Indians were persecuted, enslaved, abused, violated and yet fought on -- they won their freedom as American Indians - not as Native Americans. When you use the term American Indian - it puts the country first -- the only people to have their country first.
This to me is why Native American is demeaning - it strips away the dignity, reduces the atrocities commited and just serves to make those that did these things feel better.
In my opinion, American Indians are a Great Pround People and should be given the respect and dignity of being called the term they fought and died as -- regardless of how it makes the rest of us feel.