I am a Marine Biologist by trade, and I can assure you that not only is Aquaponics a very viable method if you have the time, money and know how to set it up, but is a completely organic nute solution as well.
The downsides are the volume of water needed to maintain the system, and the time investment you would need to find the balance point between livestock, water chemistry and plant life.
You have to take into consideration a large volume of water to house the fish, which basically will follow a 1 inch per gallon of water rule of thumb. If you need 250 inches of fish to maintain a nute level to support your plant life, than you would need at a bare minimum a 250 gallon reservoir to house just the fish, and thats not taking into consideration evaporation, and whatever the plants actually drink.
In the video posted on Aquaponics above, you are looking at probably about a 300 gallon resevoir, and if I had to guess, Id say its prolly feeding less than a 3 dozen plants, and thats if they're pushing an ebb and flow system, less for others.
To answer a few other questions:
1) Yes, you could put some guppies in your tank, or some goldfish, but yes, they will supplement their diets on vegetable matter, and the nearest supply of that is your root systems.
2) Adding flake food to the water causes PH problems as the un eaten food settles to the bottom of the tank and slowly rots. This will also cause crazy shifts in the ammonia and nitrification cycles of the water column, as different colonies of algae set up shop to feed on the rotting chemistry, namely ammonia, nitrite and nitrates, in that order. Algae in your reservoir doesn't sound good to me.
3) Most guppies, sword tails, mollies and goldfish would thrive in the reservoir, regardless of the lack of light.
This idea though interesting, would most likely fail miserably, resulting in either the death of the fish and the subsequent rotting of the animal, the food source rotting, or the animals feedding on your plants finer roots.
The uphill battle with chemistry would most likely also make this a bad nute source, unless one had available a body of water large enough to support alot of plant life, like say the size of a pond, where the unused nutes and protiens from the fish waste could be simply dispersed into the water column safely.
So in conclusion:
There is a reason the fish in the Aquaponics video are kept in a separate reservoir, and that this technology hasnt revloutionized farming across the globe.
Its simply not efficient enough to be anything more than a novelty on YouTube at this point in the game.
My advice to you is if your dead set on putting a fish in your reservoir, use a Beta, or a Siam Fighting Fish...they barely eat anything, and the wont eat vegetable matter but extremely rarely. They just sit in a corner, staring at the water surface, and hoping some blood worms fall from the sky. It wont supply shit for nutes really, but it might look cool, if you just gotta have a fish in there.
Anyway, that's my 2 pennies worth, hope it helps.
_J