The case of the Iroquois and lacrosse is really the only case I'm aware of where the issue of indigenous peoples being represented under their own nation would actually make a meaningful difference in how competitions would go.

Naturally, the Iroquois have used this to drive some more publicity for their sovereignty claims--in particular, they insist on using their passports when traveling internationally. While I'm sure the US and Canadian governments aren't happy about this, they have both ultimately said "yes, we will let them back in on these passports."

Their exclusion here appears to be because Olympic rules are that dependent nations are not allowed to to participate under their own flag, except those that had an existing committee 30 years ago. I think those rules are a mistake, and the organizers here should override those rules to let the Iroquois compete. If (perhaps when) lacrosse becomes an Olympic sport, this issue is going to become even more important, and I would hope that both the US and Canada would campaign to admit the Iroquois as nation in its own right when playing in the Olympics.