Electrolysis of aqueous solutions
It is good question given the fact that this idea is not commonly discussed. The main source of your confusion is the following idea

Why doesn't water dominate any electrolysis reaction since even if a small percentage of the collisions of water can accomplish the reaction? The sheer number of collisions on its side will ensure it wins.

Unlike chemical reactions, electrode reaction do not solely require collisions. It requires an additional process: electron transfer. No matter how much collisions are occurring, the electrode is unable to transfer or withdraw electron from water in high salt concentration. Thermodynamics tells us (from the Nernst equation) how easy it would be to remove or add an electron to a water molecule at a given salt concentration. It is silent as to why this happens and what is the mechanism.
Electrode processes are very complicated. Marcus got a Nobel Prize on electron transfer.