"As long as you take care to minimize the number and strength of the pathogens"
...but you can't. You can't control which pathogens and how strong they are. Feeding live food is not like vaccinations. It is not like if fish get ich they won't get it again. It doesn't work like that. Feeding live will simply make them less healthy, nutritionally and pathologically. If there were any increase in immunity it would be outweighed by the decrease in nutrition.
As far as them being too fat, don't overfeed. Problem fixed.
Here is the excerpt from that article about diet:
Diet is another major issue. We simply cannot simulate the natural diet of most fish. These animals have 24 hours a day, 7 days a week to forage for food. Even if I was doing research in Amazonia on native fishes, had a bunch in some research facility, was working with a native tribe and they went out and gathered food for four hours every day, I still would not be replicating their natural diet. We have no idea that although from stomach content analysis from wild caught individuals we find they eat 30% this, and 40% that, etc. that they are not also occasionally coming across other foods at a much less frequent rate, possibly with vital micronutrients or some other vital part of their diet. In addition, in the wild the foods available change throughout the year and even year to year. During the beginning of the rainy season certain terrestrial animals become a large part of the diet of many if not most fish. The rain literally washes a number of animals and other food items into the waters for the fish to gorge on. In addition to this variability, different foods are available at different parts of the day. This will be more likely to be known from observation and stomach content analysis, but there is still going to be some unknown variability here. Even our ‘natural’ diets are far from it. These diets are usually described as live foods (feeder fish, many types of worms, assorted invertebrates, etc.), frozen foods (beef heart, squid, brine shrimp, etc.), freeze-dried, and even many processed foods are marketed as natural. Some brands toss in things like fresh seafood, kelp, and other similar ingredients. The most common argument is ‘this is the type of stuff they would eat in the wild’. Neon tetras, discus, etc. do not naturally eat kelp and salmon. No fish naturally feasts on beef heart. These food items are no more natural to these fish than the ingredients in the processed foods that so many avoid because they are supposedly so unnatural. Goldfish are not found in the natural waters of Oscars. It is totally unnatural for them to eat goldfish. These foods may be similar, but it is not as natural as many may think they are.