There is no quick answer to your initial questions. Each aquarium is biologically unique, and what you put in the aquarium will have an impact on water chemistry and the biological system.
I would not advise most people to not have some type of filter on an aquarium. But the fact remains that if the aquarium is biologically established and balanced as it should be, there is no need for a filter. I had a 10g tank for over a year with no filter. But it now has a single sponge filter and I prefer that for clearer water. Note, I said clearer, not cleaner, as these are two very different things.
In any aquarium that is biologically mature/established/balanced, the substrate is the bed for the largest colonies of bacteria, not the filter. And you can do without a filter because of this. Having plants helps, even if just floating which are fastest at taking up ammonia/ammonium and organics. But the key is balanced. Most of us put far more fish into an aquarium than the system can actually support on its own. Here the filter helps, though the situation is still a problem. Never expect the filter to handle things in such situations; that is only sitting on the edge of a cliff waiting to fall over in disaster. Filters can only do so much, which is why many of us on this forum keep saying that larger filters or more filters does not have any benefit. If the tank is overstocked or maintenance is inadequate, no amount of filters is going to fix the problem.
You mention Walstad method tanks. If you read Diana's book and articles you will see that her method has some extremely important criteria from the outset. First, lots of plants. Second, minimal fish load. Most of us put way more fish in a given aquarium than Diana would ever remotely suggest.
Surface disturbance to promote a proper gas exchange (oxygen in, CO2 out) is important, but here again the number of fish is key. Plants respire the same as fish and many species of bacteria, using oxygen and producing CO2. During photosynthesis--which occurs only in daylight when (and if) the light is of sufficient intensity to drive photosynthesis--plants take up CO2, and as a byproduct of photosynthesis oxygen is released through the roots primarily and the leaves. This is fine during the day, but at night when photosynthesis ceases the CO2 is no longer being taken up, and it can easily accumulate to dangerous amounts. This is when the surface disturbance is critical, or the fish load is very low so there is no problem.
All of these aspects may have credibility, but the problem with people watching YouTube videos is that you can get the wrong idea. Some of the posters are frankly frauds as they have little if any actual intelligent knowledge of the biology/botany/chemistry. Others may have the knowledge, but they fail to properly get across the whole picture. As I pointed out above, one has to understand the whole story before diving in. Every principle of aquarium keeping may have value but none of them is stand alone. Without understanding the whole issue, running off on a tangent is not likely to work, and usually means dead fish, dying plants, and giving up.