When well established the plants will take up nitrogen as effectively as the bacteria usually used. However, initially only one or two very small fish, like one betta or two danios, can be kept and more added gradually as the plants grow and I'd recommend an airstone be used for circulation. Although a plant-only unfiltered tank is possible, even a small filter in addition to the plants would be helpful in case the plants die or something so you have back up biological filtration.
Your idea isn't wierd. Terraphyte tanks have been around for quite a while. I use land plants to filter the tank water. The system works fine and the fish have pure water. You can use standard filtratation to supplement the plants' work and filter the tank water at night when the plants slow their filtration.
Tank is pretty much maintenance free. No ammonia or nitrites in the water and the nitrates are very low. Few, if any water changes are needed, because the plants maintain the tank water.
Your basic concept is quite valid. Plants will indeed remove a lot of the waste from an aquarium.
The most basic concept is ANY sort of circulation, then put plants in the path of that circulation. There are all sorts of ways of setting it up.
In a pond set up I have seen the same idea. It is quite common.
Set up 2 ponds. One is the main pond. The other is more of a shallow stream. Put lots of plants in the shallow stream (in pots or planted directly in the stream). Grow the plants that thrive with their leaves out of the water. Pump the pond water through the stream. Stream is about 10% or more of the volume of the pond.
For aquariums I have seen quite a few ideas.
Planted sump. Refugium.
Set up a box above the tank for the plants and pump some aquarium water through that. The box is planted with house plants, vegetables or other plants.
Look up Aquaponics, Hydroponics and related terms. These deal more with growing vegetables, and some incorporate raising food fish with the vegetable gardening.
You can get as complex as you want, or keep it as simple as you want.