The largest problem is that both names fly in the face of how actual tribal members identify themselves. Both names have a common component, "American." But the tribes identify themselves individually as Cheyenne or Apache, not commonly based on continental grouping. The American grouping is an artificial construct of the Europeans. The name itself is based on that of Italian explorer Amerigo Vespucci.
Why would one name bestowed by an Italian explorer be better than another name by a different Italian explorer?
The very construct of a name for everyone who is descended from people living in the Americas in 1491 is a very eurocentric approach. France and Germany are adjacent to each other and no one would characterize the French by German actions in, say, 1942. But people have no problem lumping together the diverse tribes under the Native American label. That label would of course fit just as well for the hundreds of millions of people who were born in the Americas, whose immediate ancestors were born in the Americas, and have no connection with relatives elsewhere.
Why would tribal members prefer a label that lumps them with the descendants of their oppressors over a label that lumps them with other victims? Neither is really accurate. Neither is based in their own words.