Azndragon: It is a bit of nostagia to see that Walstad article. The science is good. When I got started with plant aquaria in '91-'92, Diana was my hero(ine). Solid on the science, but also an advocate of low-tech aquaria techniques. I loved the combo & ditched my CO2 infusion. We corresponded (no email in those days!), and her advice was usually on target even when she had to be intuitive to fill the data gaps. The part of the puzzle you're missing, however, is the volume of ammonia emitted by the fish and the quantity/growth rate of plants required to utilize that volume of ammonia. In the plant aquaria hobby, a 50 gallon heavily planted tank might have a half dozen Otocinclus, a few mystery snails and a dozen cardinal tetras. Forget the pics in the Amano books with two dozen tetras, and a dozen shrimp in a 15 gallon tank. They were show for the photos. Look close & you'll see that the tetras were tiny specimens, freshly imported and a bit scrawny.
The volume of actively growing plants necessary to consume the ammonia emitted by one 24" koi would fill the surface of a pond having several hundred square feet of surface area. Remember, in Nature they live together... hundreds of thousands of gallons of water filled with plants and one carp. For plant aquaria, the fish are secondary. They add some movement; and their ammonia and other wastes fertilize the plants. But, if there are more than a very low stocking, there is too much nutrient and algae will overwhelm the tank.
There is an old thread where I ranted about plant filter hype hoodwinking consumers. Search for it. Worth the read!
BTW, I still have my plant tanks...and I keep a lot of fish in them. Long story of what it takes, but I don't rely on a "natural balance".