Maybe it's not clear in the movie, but the Guarani (the indigenous peoples) missions freaked out the colonial powers (Spain & Portugal) because they lived in them as Guarani, NOT as colonized peoples with their former culture erased.
Yes, the Jesuits converted them to Christianity, but the Guarani kept their language, culture, societies and their identity as Guarani intact.
You would have to think back to the Collapse of the Roman Empire/early Dark Ages in Europe to find something similar, where different societies (the Irish, Germanic tribes, etc.) abandoned their traditional "pagan" religions and joined "Christendom".
Needless to say, such a thing never happened again. The events of The Mission are part of what got the Jesuits kicked out of the Americas in 1767.
For some more details, here is a decent Wikipedia article (it's late so this is what I found first): https://en.wikipedia.org/wiki/Jesuit_reduction