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.
while everything you said makes sense, the thing I keep coming back to over and over again is that the oldest human remains in the Americas have always been in South America. Which to me has always implied that the first people arrived in South America from somewhere, and the closest possible racial match is the Polynesians.
The Polynesians could have come from What is now Australia and nearby to what is now S America-it is a long trek, but they were skilled sailors since day one. Also, the sea level in the late ice age was a lot lower-maybe allowing for stops to get provisions before sailing/paddling on. However it happened, they managed to get there, bringing their DNA early on-I’d love to know who-if anyone-was already there in what is now the far edge of SA for the sailors to interact with-the whole concept is fascinating...