Hello; Bettas can survive in very small containers largely, I think, because they can hold a small air bubble in their body and get supplemental oxygen. Fish without this adaptation would use up the O2 in such a small amount of water. I also think that the native area for bettas can have high water temps which leads to reduced O2 levels. Bettas are adapted to low O2 saturation.
Beyond the O2 it seems to me that the other aspects of water quality should apply. Metabolism byproducts will show up in the water. Decay byproducts from uneaten food and such should be in the water. One rule of thumb I have gone by for years is to have the largest tank that is practical as I have found the larger volume to be easier to maintain. Put another way, smaller tanks can go into some sort of crisis quicker that larger ones. The same sort of rule can be applied to stocking density. I use the analogy of a large tank with low stocking density being like a wide road with lots of buffer capacity before going into the ditch. A small tank or one with heavy stocking like being on a tightrope that will put the tank into a crisis with small problems.
I like to keep a betta in five gallon tank, but can see as small as one gallon being workable if you stay on top of the water conditions.
Breeders have a particular set of circumstances. They must raise a lot of individuals and the males cannot be kept together.
One way is to use small containers and have a lot of them. To me this must require lots of attention to the water changes and all that involves on a daily basis. It also likely requires the general environment of the room needs to be kept around 80 degrees.
Another way is to partition a tank so that the bettas are kept apart but the larger tank is filtered and heated like any other tank. The partitions allow for a flow of the water. In the past such tanks with betta chambers were for sale in fish stores, I have not checked in years.
In addition to a five gallon tank, I like to keep the water temps around 80 F for bettas. I also think it is better to have a tighter cover on the tanks for maintaining a higher humidity in the air above the water. My thinking being that as the bettas take a gulp of air, warm moist air is better than cold dry air.