Bird malaria is one of the worst causes, but there are many others. By the 1920s, so few native birds were found in the coastal towns that well-meaning people brought in tropical species to fill the void. Other birds were introduced to help control insect pests in the pervasive sugar cane fields. One can see the point of choosing non-native birds versus no birds, and one can also realize the lack of research that would have warned against introducing new species without understanding the potential negative effects.
In any case, they are on the islands to stay, for better or worse. And they may be filling the environmental niche left behind by the sadly extinct native birds. And if you think about it, the most populous bird on the island could have been something far more obnoxious (I think of European Starlings on the mainland, for instance).