It is difficult for me to believe that Polynesians reached South America in sufficient numbers to make a DNA impact.
They literally had to cross thousands of miles of open ocean.
Bad luck in timing meant no rain fall for water and no migrating fish for food. Few islands meant no wood for repairs or replacement parts. I am thinking the death toll for Polynesian explorers, in a good year, was 90%.
During the glacial era, there were probably a lot more southeast Pacific islands than there are now.
On the other hand, ancient South American art does not indicate a common tradition of long distance ocean crossing.
As z implies, the Polynesians spread out as small groups, skipping a number of atolls and such which were barely above sea level. Those tiny bits of land only had some temporary populations during the European colonial period if there was an economic reason, or a supply of fresh water and supplies for various (mostly British and French) navies -- or during the WWII yearss, and most of those were designated bird sanctuaries and whatnot after the military expediency passed.
Also, regarding the Polynesians, their arrival in Hawaii didn't happen until the late 12th century at the earliest, and a bunch of civilizations of South America had risen and fallen a bunch of times by then. The Polynesian population on Easter Island arrived sometime in the 13th century to a place already occupied by some kind of culture from South America.