Short answer Squirrel is your half right, it is genetics, and the rest is culture.
Of those nations: Tonga, The Cook Islands and Niue are all Polynesian by ethnicity with Nauru being a mixture of Polynesian, Micronesian and Melanesian and Micronesia being majority Micronesian. Its a common practice in Melanesian and Polynesian culture to measure wealth by weight, Tonga, the Cook Islands and Niue all practice this to a varying degree; with Tonga being the standout (the former King of Tonga weighed around 300kgs), with the Cook Islands and Niue practising much the same on a slightly smaller scale. Nauru is heavily influenced by neighbouring Polynesian states, such that it has adopted similar cultural practices, including correlating weight with wealth. Likewise Im led to believe that Micronesia has similar practices and a much richer diet than the others.
The main sources of food on all those islands are Taro, Pigs (dogs as well), Sweet Potato, seafood (in many islands this was reserved for special days and the aristocracy) and other tropical fruits and vegetables. Taro which is the staple food, is extremely high in starch (sugars) which combined with the other remaining fare is very high in fat, sugars. Quite simply these are all fattening, imported high fat foods are not really all that different from what is already eaten natively.
I wouldnt denigrate Nauru, its was and is a veritable paradise, you dont need to spend much time working to gain a meal, and you dont work much there anyway, the day starts early for Church (depending on faithfulness), the morning is when you get the bulk of your work done, nobody does anything during the hot afternoon and work begins again as it cools down. I will also note that Phosphate mining from there was started by the Germans before the First World War (Nauru was part of the Marshall Islands Protectorate of the German Empire). And from 1970, Nauru controlled the Nauru Phosphate Company; they were also one of the richest nations in the world per capita, comparable to the Oil Sheiks and Gulf States. Aside from slightly higher unemployment, Nauru isnt really all that different from many other island nations in Oceania
negotiated with effectively tribal societies, and thus the entire populace ended up benefiting
No, the Tribal Chiefs in many of those islands are corrupt, and the money which is meant to work it's way down really just settles into someones pocket (be it the governments, normally run by the chiefs, or be in your chiefs pockets). And you do not ever object to such conduct.