I am heading out on an errand, so here is something I just grabbed to respond. The first "no" concerns the disease issue, but what follows is along the lines of what Kyle mentioned, nutritional issues:
A second disadvantage is nutritional imbalance. Goldfish in particular are fatty and are not a healthy staple diet for most piscivorous fish. In the wild, piscivorous fishes will take a wide range of species, some herbivorous, others smaller predators. This means that the piscivorous fish will be able to obtain a correspondingly wide range of nutrients. Simply feeding one species of feeder fish is both unnatural and very likely unhealthy. This problem can, to a degree, be mitigated by "gut loading" feeder fish. By contrast, flake and pellet foods have been carefully formulated to provide a perfect diet for fish. While it might seem monotonous to us, these prepared foods are actually the best all-round diet for most fish.
A third disadvantage is that some feeder fish (notably goldfish and rosy-red minnows) contain large amounts of the enzyme thiaminase. This breaks down thiamin (vitamin B1) and over time this will lead to serious health problems.
Back to Dave's observation on this being "natural" in the wild. The fact is, that it is anything but natural for most of our aquarium fish to eat other fish. The prime foods taken by characins, cyprinids, catfish and so on are crustaceans, insect larvae, worms, insects on the surface, and plant matter in some cases. Fish-predators are usually the much larger fish, and here they do not stay with one species. And as it mentions above, not all fish are made the same so there is variety.
On the angelfish, it is true that it is an opportunistic predator, meaning, it will predate small fish if given the opportunity, but this is not "normal diet" in its habitat. Wild angelfish rarely eat other fish.
Byron.