Why doesn't separating the layers change the equilibrium point? I'm following this tutorial for synthesizing esters. I don't understand why adding water to the separation funnel doesn't push the equilibrium reaction point back to the left, or why draining it doesn't push the reaction to the right, given the tutorial's explanation of Le Chatelier's Principle:

When the reaction reaches equilibrium there is still a large amount of reactants left in the mixture resulting in a poor yield of the ester.
The yield of ester can be improved by increasing the concentration of one of the reactants (either the alcohol or the carboxylic acid).
By Le Chatelier's Principle an excess of one reactant will drive the reaction to the right, increasing the production of ester, and therefore increasing the yield of ester.
In our experiment we are using an excess of carboxylic aicd (acetic acid which is also known as ethanoic acid).
From the balanced chemical equation, you will also appreciate that the presence of water in the reaction mixture will drive the equilibrium to the left, favouring the formation of reactants rather than ester.
Draining the aqueous phase cannot change anything. If the phases are in equillibrium before (both internally and with each other), they stay that way if they are separated.
Now if you add water to the aqueous phase, the reaction in the unpolar phase (which is saturated with water, as the phases are in equillibrium) is in the first moment not affected.
However the acid, alcohol and ester have different solubilities in the aqueous phase. That means the concentrations in the unpolar phase will change a bit, and so does the reaction equillibrium.
But: This is all irrelevant, because the reaction happens under reflux. While you are standing there with the cooled mixture in the separation funnel, the reaction rate is much lower, you will never notice a change.