I wouldn’t describe it as “winning” or “losing”. In the US, Canada, and Australia, the white majority rried to eradicate any non-conforming cultures, whereas in Latin America, Africa, and India, the white settlers in power were so drastically outnumbered that they used various forms of racial hierarchies amd segregation. When those colonial empires collapsed the governments became more representative of the local populations. They still oppressed (and continue to do so) various groups, but indigenous and historical cultures were able to survive due to large populations that were able to carry on those traditions.
Brazil actually merged the freed slaves into society because people at the time thought that over several generations, everybody would end up being white again. In a different way they were also trying to suppress them.
As for the indigenous population, before Portugal arrived here there was one large tribe already dominating all the others. The Portuguese then negotiated with that large tribe and that one tribe’s culture managed to survive, but the colonizers also had no respect for it or any of the others and grouped them all together as if being the same thing. The other cultures ended up being either absorbed or erased by that larger tribe.
Do you think people would feel better about how america handles culture if they would stop replacing their culture with things like shopping malls and business center?
Maybe the problem is more about Americans destroying culture and not replacing it with anything that will last or represents them.