There is no true self-sufficient aquarium system.
If you understand aquarium nitrogen cycle, you should know fish produce ammonia. Ammonia is toxic to the fish, it must remain at 0ppm concentration in the aquarium water or the fish will eventually die. In well established aquariums, filter systems can develop colonies of specific bacteria species to feed on ammonia. They turn ammonia to nitrite, before turning nitrite to nitrate. Although nitrate is relatively safe, too high concentration of nitrate is still bad for fish. Now there is no true way to get rid of nitrate from a closed system such as in an aquarium. It is why we often do weekly partial water change. The main purpose is to get rid of nitrate.
It is true aquatic plants can absorb a small amount of ammonia and nitrate. The quantity is too insignificant to be enough. You will need to have plants covering every square inch of a 30 gallon tank just to be able to take care of the ammonia produced by a single 1.5" Neon Tetra.
I have been running planted aquariums for years and I know how quickly nitrate can build up even in heavily planted fish tanks. Unless we are talking about just one tiny 2" fish in a 55 gallon heavily planted tank, you will still need an aquarium filter system to convert ammonia to nitrite or there will be ammonia spike. You will still need partial water change on weekly basis to keep nitrate down.
Now what you talked about is similar to a sump. It is usually a secondary tank hanging behind the main show tank, where you can add additional filtration to it without making your show tank look bad. Certainly you can plant tons of plants in the sump if you wish, but it won't be able to take care of the ammonia produce by the fish unless we are talking about a 5,000 gallon sump for a 50 gallon show tank.
The function of a sump is usually to hid your filtration system from the show tank, and of course it will allow you the option to have more filtration than just have a normal tank without sump. It will help the water quality for sure but not to the point where you do not need to do partial water change.
You also need to understand that nitrate is not the only organic waste ended up in the aquarium. There are also phosphate and other stuff produced from rotting fish waste. Without partial water change once in a while, the fish will be literally swimming in their own waste. It is unhealthy for them.
