I started to look up links for this post, but honestly, I'm just to tired and lazy at the moment.
While there might not be the same level of plankton murking up the water that is seen up north, there is still plenty of plankton in those "clear" tropical waters. Lot's of larval fish, and other larval critters like polychaetes, not to mention copepods.
Terms like "plenty" and "lot's" are relative. If we're comparing tropical coral reefs to oceanic deserts, then you're right. If we're comparing them to the vast majority of the rest of the ocean, then they are nutrient poor environments.
Larvae of many of those critters don't survive for long, if they stay near the reef. There's simply not enough nutrients in the water to sustain them. One of the examples you brought up was fish larvae. The majority of these fish larvae that are lucky enough to survive, do so by moving to more nutrient rich habitats, like grass beds and mangrove swamps. That's why these areas are referred to as fish nurseries. They don't survive by hanging out on the reef.
While it's certainly true that corals need their zoox...that just supplies energy (basicall carbs), they still need more nutrition than that. Hence all those polyps, think hungry mouths
They wouldn't waste the energy growing those polyps and the nematocysts if there wasn't food to catch.
It's been shown in several corals that were studied, actually digest the zooxanthellae themselves. This means they obtain more than just carbohydrates from their zooxanthellae. They also obtain nutrients like phosphate, nitrogen, and amino acids from the zooxanthellae cell walls.
Corals use their tentacles, nematocysts, and mesenterial filaments, in territorial disputes. Their gastrovascular cavity is used in reproduction and zooxanthellae population regulation. Growing these structures wouldn't be a waste of energy, even if they never used them for prey capture or feeding. The fact that they have these structures is not proof that the coral is actively feeding. Ostriches have wings, but we'd be wrong to assume they were used for flight, simply because they have them.
With that said, I do agree that corals feed. The corals dependence on prey capture would vary depending on species. With virtually all zooxanthellae containing corals, they can not survive on prey capture alone. When a section of a reef bleaches, there's only two options. Regain the lost zooxanthellae, or starve, because there isn't enough nutrients in the water to sustain them. Either dissolved, or in plankton.
Humpback whales come to these clean waters around tropical coral reefs to nurse their young. The whole time they're there, they don't feed at all. Why? Because there isn't enough food in the water for them to feed.
Lots of plankton can't be maintained without lots of dissolved nutrients to sustain them. There can be temporary plankton blooms without high levels of dissolved nutrients, like mass coral spawning events, but these are temporary. In order to maintain lots of plankton, there has to be lots of dissolved nutrients. We can't say that reefs have little in the way of dissolved nutrients, but lots of plankton. This would be like saying there are lots of lions in Africa, but very few wildebeests. It just doesn't work that way.
