How was the idea of being exiled both alienating and enriching to American Indian culture?
Let's examine the second part of the question -- how was the idea of being exiled enriching to the cultures of Native American tribes -- first, as that is quite simple and leads logically to the first part of the question -- how was the idea of being exiled alienating to Native American culture.
The violent and protracted processes involved in forcibly exiling indigenous populations from their traditional lands while simultaneously pursuing genocidal practices intended to force assimilation into European cultures was in no way enriching to the tribes affected. The legacy of these processes is evident today in alarmingly high rates of drug and alcohol abuse and domestic violence among many Native American communities. Indigenous tribes were decimated and forced onto reservations with far worse sustainability characteristics then the land from which they were exiled.
The Native American experience dating at least to the arrival of 15th and 16th century Europeans, including Christopher Columbus and Hernan Cortez, was entirely negative. Europeans brought, both accidentally and deliberately, diseases alien to indigenous populations who lacked the antibodies needed to fight of infection. Europeans used deceptive negotiating practices that tricked indigenous tribes out of their natural resources and land. European, and later American settlers and armies also forcibly drove indigenous tribes off of their land and deprived them of access to their traditional hunting grounds. As recently as the middle of the 20th century, Americans have continued to attempt to force native tribes to assimilate by depriving them of the freedom to use their own languages and to retain their own unique cultures.
Basically, there was no enriching experience for Native Americans associated with exile. It was all disadvantageous by design.