I don't know if the author of the article misunderstands or misrepresented the theorem, or if the theorem is flawed. Anyway,
The incoming and outgoing flow of water ultimately determines the water level and stability of the bathtub.
The incoming and outgoing flows don't determine the level, they determine the rate of change in the level. For example, if the incoming flow is 2 gallons per minute and the outgoing flow is 1 gallon per minute, then the level of the water in the bathtub will rise at the rate of 1 gallon per minute, regardless of the level.
