Its an interesting contrast between the coral biology community and the hobby on this one.
In nature, its important for corals to feed. While the metabolic contributions of photosynthesis are important, and people looove to cite the paper that says up to 90% of a corals energy requirements *can* be met by photosynthesis (in some circumstances), the problem is that this is entirely carbohydrates. Coral biologists speaking informally about this at conferences refer to it as junk food - pure sugar. In nature, corals eat constantly, and need to.
Funny thing is how little effect feeding has on corals in our tanks. Yes there are some papers showing benefits of some foods for some corals. But the effects from published aquarium coral feeding studies are pretty underwhelming overall IMO. (Not a criticism of the experiments at all - I'm saying that I'm surprised the effects werent larger)
Personally I think it comes down to food quality. In nature they eat live zooplankton 24-7, then we put them in tanks that in relative terms contain almost no zooplankton, and try to feed them things like processed fish meal. Small wonder we end up debating whether feeding even matters