You won't an arm of the US Government invade an embassy we want to continue to have relations with any time soon.
Foreign embassies are totally different. Besides, the US government was forcibly removing native children from their parents without cause into the 1970s. You can't get much more invasive than that.
The point I am making with the "public land" comment is internal. It isn't that the US gov't owns the land, it is that the "tribe" owns the land. Not the individual living in a house on the land. How can you say a police officer doesn't have the right to search your private property when it isn't your private property? I;m not sure how the leases work and haven't found much information on it, but it certainly seems like the "tribe" effectively owns all real property.