My understanding about the problem is buried down in the article-
Besides the damage inflicted by sea-level rise itself—coastal erosion, surface flooding, and saltwater intrusion into soil and groundwater—they will suffer from increasingly frequent and severe weather extremes (droughts and cyclones) and die-offs of their coral reefs through ocean warming and acidification, leading to potential collapse of marine ecosystems that provide food and livelihoods for island dwellers.
A decade ago (geez!) when I visited the Cook Islands they told us the issue was not them literally sinking into the ocean (Rarotonga, the main island, is a giant volcano and is not going anywhere) but more what's written here. Cyclones in particular are the awful ones- beyond the devastation a direct hit can cause, floods from them can make the water and arable land brackish, and you basically can't recover from that. And when you're only a square mile wide or two, this is a huge deal and you can't just move inland for more arable land. And when the salt gets into your drinking water, well, you're fucked.
Coral reefs are the second one- most of these islands still really rely on fishing for a large percentage of the food intake (plus money from tourism and all that fun stuff). When the coral reefs die, sustainability becomes a huge problem.
But yeah, I mean, there are definitely some islands that are going to disappear. But like most places on Earth, the risk is not from the slow onslaught but rather the extreme events- even in the Netherlands no one expects the flooding that overwhelms the dikes to come from the ocean, but rather from a sudden deluge of water from the Rhine after a flood.