"Quicker" isn't quite the right word, because the prediction says nothing about the dynamic situation. The equilibrium is very fast, however. You can think about it this way... every ammonia molecule is constantly acquiring an H+ ion (becoming an ammonium molecule) and then releasing it later... and repeating that process over and over and over... the equilibrium prediction tell you how many of the ammonia molecules are ammonium at any given time... it says nothing about how quickly equilibrium will be reached, though, again, in this case it is very fast -- fast enough to assume that it is instantaneous.
There are equilibriums that are very, very slow to reach equilibrium. For example, put hydrogen gas and oxygen gas into a sealed vessel kept at 25 degrees C. The equilibrium state for that is water, but you can open that vessel up 50 years later and there will no significant water vapor or liquid water. Using the energy measures that show how far from equilibrium a situation is, hydrogen gas and oxygen gas is energetically very far from equilibrium with water, but, just because it is very energetically favored, it says nothing about the speed at which equilibrium will be achieved Conversely, energetically, there is not a large difference between ammonia and ammonium, but the equilibrium process happens very fast.
The equilibrium dynamics say that the lower the pH, the more the ammonia-ammonium balance will be shifted to ammonium. But, it is important to say again, that there will always be some ammonia and always some ammonium. That's what the calculations in the linked article show. There will never be a situation where 100% of either will ever be present (without something artificial occurring). There will always be some of one and some of the other, just how much of each is a function of pH, of temperature, of the other things dissolved in the water (e.g. salinity).