I do try to have plants in all my tanks (where they won’t get eaten). No matter what, you need some sort of filter or bubbler to turnover the water and aerate the tank though.
In one of my tanks, the filter is only used for turnover and extra space for bacteria, so I suppose that counts as the plants (or algae) being the main source of filtration for nitrate - must be working for nitrates to always come back undetectable.
This only works in a lightly stocked tank though - plants can only keep up with so much.
Agreed...with one proviso. While filtration and aeration are undeniably beneficial and make aquariums easier to keep...it's simply not true to state that they are essential.
A planted tank can absolutely support fish with no added filtration, aeration or any other water management/movement. This is best attempted with fish from slow-moving or stagnant waters, rather than riverine species...and of prime importance is the fact that these aquariums cannot begin to support the level of overstocking that almost all aquarists insist upon these days. But if you are willing to accept sparsely-populated tanks and generally small fish, a quiet planted tank is not only possible but surprisingly easy.
I kept fish for several years as a young kid (with plenty of supervision and help from my father) and even though simple vibrator-type air pumps were available, I didn't have one throughout those first years. It can be done...it was done as a matter of course for many years in the infancy of the hobby, and our understanding of the processes that take place in underwater environments makes it even easier today than ever to be successful with the super-low-tech or no-tech approach.
But small air pumps and sponge filters are so inexpensive these days that the only reason to try this is because you want to try it, not because there are any advantages to it. It's cool and satisfying to do it successfully; but it's like going grocery shopping in your classic Triumph Spitfire rather than the family minivan...fun, but not measurably better in any other way.