Very interesting idea.
For example, let's say that you fed the fish 4% of their daily food intake every hour and also had 4-5% of the water being dripped out every hour. Ammonia buildup in the tank would be constant after a point, and a small amount of BB would exist all over the tank to handle nitrogen conversion. If there were some plants growing in the tank, and some amount of purigien and a reasonable water circulation, I think there would be a chance you could get by with no added filtration.
However, people don't feed that way. Ammonia is not dispersed after feeding in one level amount for 24 hours: most is disbursed within a few hours. So, it will peak to dangerous or lethal levels with even an outflow of 4-5% within hours of feeding. Moreover, a lot of people won't remove water at that rate (96-120% per day.)
One could get by this way on fish that are non-stop browser types (herbivorous or sand sifters) or maybe if the feeding was set up as a small fraction each hour. Or obviously if a lot of water replaced each hour (40-50%.) That's what nature does basically along with keeping stocking levels far, far lower than in artificial environments. All the fish don't eat at the same time in a river and overall there aren't that many fish.
The most critical issue, imo, is the pattern of feeding people normally use. It's like a grocery store: if the only time people went to the grocery store was either 8 AM or 7 PM, every three days, there would be a disaster. There would not be enough registers, not enough employees to keep stock on the shelf, not enough parking spaces, not enough space for grocery baskets, so not enough baskets. And if the store hired more people they would have nothing to do between 10 AM and 7 PM so the employees would have to work a split shift or the store would lose money. It only "works" because people happen to use the store almost all day long every day.