No, what got the NCAA off FSU's back was that the Executive Committee implemented the policy believing the Seminole tribe in Oklahoma was against the FSU nickname. The NCAA knew full well that FSU had the support of the Florida Seminole tribe, but it counted on opposition from Oklahoma to justify its action. But the Seminole tribal council in Okahoma almost immediately passed a resolution overwhelmingly supporting FSU, making the NCAA's claim that it spent more than four years exhaustively studying the issue look as bogus as it was. The namesake tribe exemption had to be invented out of thin air to enable the NCAA to save face.
I read that there's at least one Ute tribe in Colorado that opposes Utah's use of the Ute nickname. I know for a fact that there are other Chippewa tribes that oppose Central Michigan's nickname. UND has a resolution from the Spirit Lake Sioux tribe supporting its use of the Fighting Sioux nickname. Using the NCAA's own standards, that should be enough. What Sioux tribes in other states think shouldn't matter.