When Geronimo was running loose, Americans considered him a vicious killer. But once they caught him, their attitudes changed. The following quotes tell the tale:
What had changed was America itself. Geronimo’s surrender had ended the Indian Wars that had raged for nearly three centuries. (Narration)
He becomes an icon, a sentimental icon, of what was once a real enemy. And there’s something amazingly American about that transformation. (Historian)
1) Indians joined our culture (albeit unwillingly). They're part of us now. We've proved ourselves superior, so they saw the error of their ways. Now they're civilized instead of savages, just like us.
2) Indians were fierce and warlike--a "challenging, dangerous" foe. We had to be even stronger and tougher to defeat them. But we did it. Our victory over these "honored warriors" proves we're the greatest warriors of all.
As I said in Smashing People:  :The "Honor" of Being an Athlete, if we really wanted to honor ferocious fighters, we'd honor those who almost defeated us: the Nazi blitzkreig, the Viet Cong, the jihadi terrorists. But these warriors showcased our weaknesses as well as our strengths. We didn't get to parade them to show how much better we were than them.
In other words, they punctured the myth of America's invincibility. They demonstrated that determined foes can beat the American fighting machine, at least temporarily. Hence they can't be our mascots. They're not tough in a primitive, outdated sort of way like our quaint li'l Indian savages.
Think of the macho type of man who walks down the street with a Doberman or pit bull on a leash. This is exactly the mentality embodied in an Indian mascot. "I have a mighty dog on my leash" is the same as "I have a mighty Indian on my logo." They're both ferocious creatures, but we tamed them. Now they belong to us.
For more on the subject, see Apache Views in Geronimo and Review of Geronimo.
Below: "Honoring" Indians as noble losers.