Is a very informative thread and I will humbly enter my two cents worth.
I have been keeping fish since I was sixteen and am 53 years young at present. Fishes will always do better in properly heated,cycled,and filtered tanks and Bettas are no different. Five gallons would be the minimum in my view. The problems with smaller tanks are numerous . Poor dilution of waste capabilities between water changes,fluctuating temperatures that are more easily controlled in heated tanks with larger water volume ,Uncycled tanks readily allow toxins such as ammonia and nitrites to affect the fish in negative way due to possible over feeding and waste created by ALL fishes unless water changes occur with proper frequency which could mean daily, depending on volume of water. (Bigger the tank, the more dilution capabilities).
Those who keep Bettas in small, unfiltered,uncycled vases,bowls,jars,or tanks smaller than 2.5 gallons aren't doing the fishes any favors and any book written on proper care of tropical fishes will tell you so.
Fish create ammonia each and every day through respiration, (breathing) and through waste created (poop). without a properly cycled filter,, they are swimming in toxic condtions between water changes on daily basis= sick bettas. This will ALWAYS be the case.
When I began keeping fish, I single handedly murdered untold numbers of fishes through ignorance on the proper care of them. With the information published in books today, and available on the internet,, it passes my understanding as to why ,,,people continue to kill their fishes.
Bettas and goldfish are the most abused fishes in the trade ,and when people finally grasp that 80 to 90 perecent of ALL fish health issues are directly related to water quality ,or lack thereof,, then perhaps providing them with the enviornment needed to ensure their health will become the norm.
But I ain't holding my breath.