Was the study of this island controlled for climate, compared to a nearby island with different diets, or compared to a distant island with similar diets?

One of the points made by Ioannidis is that we’re far too easily swayed to believe what sounds plausible when no one has yet asked “Could this be for some other reason altogether that we aren’t testing for?” and then tests for those reasons. That flaw applies here: studying just one community automatically implies that the results are invalid, simply because there’s no evaluation of other communities that share various attributes that could be hidden factors for the benefit.

EDIT: Per one article, that study “would try to cut through the stories and establish the facts about Ikaria’s longevity”. It has succeeded in establishing that, factually, Ikaria’s longevity is a thing relative to other populations. It has not established which of the properties of Ikaria are responsible. (For example, perhaps the island is slightly radioactive, and old people thrive in a slightly higher radioactivity.) Extrapolating that their diet is responsible is an instance of the A/X error described in the above-linked article.