Palaeo-anthropology has come a long way since Heyerdahl’s day. There’s not much doubt that Polynesians are an mixed “race” with Austronesian origins and a strong admixture of Mongoloid Asian and various other groups. This is borne out by genetic, linguistic and archaeological evidence.
In horribly oversimplified terms, the black skinned people who originally lived from southern India through to Australia developed a seafaring, coastal culture, most likely on Taiwan. That seafaring trait meant that they managed to survive in the face of Mongoloid agricultural expansion when other Australoid people in Asia either went into decline or became swamped by Mongoloid expansion. It also, of course, led to them moving around a lot and in the process they picked up genetic and cultural material from the agricultural peoples of China and New Guinea. So you’re right, they are vaguely Asian, being basically mixture of SE Asian Australoid and Chinese mongoloid.
Over time one branch of the now thoroughly mixed seafarers, the Lapitans, developed into what we know as Polynesians today.
Polynesians certainly never came from South America. There is some evidence that proto-Polynesians may have *been *the original Americans, traveling up the coast of China and and across to North America and thence down the coast. Not a popular theory, but held by some mainstream Anthropologists. But if that is the case then that means that the Americans came from the same stock as modern Polynesians not that Polynesians came from the Americas.
There is also some evidence that Polynesians traveled to South America and back again, leaving chickens in South America and bringing sweet potatoes and other foods back to Polynesia. This is far from established but once again it’s accepted by mainstream anthropologists. And once again, this is not in any way the same as Polynesians coming from South America. Polynesians went to South America, they went back to Polynesia. They may have taken some people with them from the America to Polynesia, but it was such a small number that they left no detectable trace in their genes.