I've never set up an aquaponics system (only discovered it last week) before but i'm interested in the systems.
I used to keep a few aquariums when i was younger. Had a couple of 4 foot tanks running with two undergravel filters in them. Obviously the fish waste accumulates under the filter and all is needed is an air pump creating a vacuum , which causes the suction of fish waste. I would think its quite an efficient system as most fish waste drops to the bottom of the tank anyway. You would not be trying to collect the waste from the entire tank, but just from under the filter.
So would it work if one was able to collect the waste under the many filters by pipe(s) in the bottom of the fish tank? eg you pump or drain out the accumulated waste once a day thats under the filters for a couple of seconds/minutes onto the plant bed ?
If so would you need a light continuous spray of water onto the plant bed all day or would the couple of minutes pumping of even distribution be enough ?
Actually, the way an undergravel filter works is to trap the waste in the gravel itself where the majority of nitrifying bacteria live. The stuff you see is below the filter plate is excess that has dropped or been pulled through. If you use airstones in you uplift tubes you will see a fairly clear area near the tube and more crap further away. If you used a powerhead the water would flow faster and less would settle under that plates.
I that same aquarium gravel size made a tremendous difference. Fairly fine gravel (like coarse blasting sand) would filter better than most. Too fine and you get compacting and water flow restriction. Too large and the water and waste flowed past and the water is hard to clear. So pebbles would not filter well on it's own but mixed with the finer gravel would look more like a natural river bottom...
That being said, trying to rinse that gravel and sending it to a planting system would be a horrendously labor intensive proposition.
Water flow in the aquarium is important in an undergravel to not only support the fish requirements but oxygenate the water all the way through the filter.
Now in an aquaponics situation, a multitude of media sizes can be used and be efficient. For a couple reasons. The main one is relatively high oxygenation rates like with a flood and drain system where water and air exchange places. The other factor is simply water flow..exposing as much water as possible to the available substrate.
It gets involved, but you can look up the oxygen dissociation curve where you can see how temp, exposure to air and liquid flow all effect the rate of exchange.
In simple terms using your growing beds, piping, and the water itself as the method of oxygenation and denitrification, and at the same time using them to serve as a grow media is the better choice. The trick and I am a beginner here is getting that mix to serve your needs and fit your space and budget and still produce the quantity you want.