The original piece does address this. The gist is that the volcanic plateau underlying the carbonate platform influenced the early morphology of the platform (duh) but that its influence became "muted" once the plateau was entirely underwater. Volcanic activity is unlikely to be related beyond that initial effect.

I was wondering the same. Evidence for this would be atolls having formed around volcanoes prior to the rapid fluctuations in sea level, but according to this article, at that time corals formed flat-topped reefs.

Decades ago, Hans Hass noticed that some atolls did not fit Darwin's model, and proposed that these formed because of better nutrient flow around the edges, but the prevalence of these flat-topped reefs suggests that is not the main story, either.