Using any people or ethnicity as a mascot is not acceptable. It objectifies the culture and the people within it. Think about a sports team called the London Chinese, with a Chinese person being their mascot. It doesn’t matter how tasteful the logo or the mascot is, the fact that it is objectifying an ethnicity in this way is offensive.
Also, offence is subjective. In the example provided of the Florida State Seminoles, it is true that the members of the Seminole tribe living in Florida approved the use of their people as a mascot for the football team. However, members of the tribe living in Oklahoma have not condoned the use.
If even one Native American finds the practice offensive and unacceptable, then it is. It is offensive because it causes offence.