These analysis are logically correct, if the second equilibrium is much stronger than the first equilibrium. That is, dilution affects the second equation much more than the first equation.
When water is added, the immediate changes are all aqueous species have a lower concentration. This implies the first equation will move to the right to increase [Ag+] and [Cl-], and second equation will move to the left to increase [Ag+] and [NH3]. If the second equilibrium is much stronger than the first, [Ag+] will be increased by a significant amount, and thus pushing the first equilibrium to the left. But this occurs only if the maths agrees with 'second equilibrium is much stronger than the first'.
My initial calculations are showing that this is not the case. This system is not easily solved analytically, I am numerically approximating equilibrium concentrations and then altering the system after equilibrium is achieved. What I have found is that diluting this system already in equilibrium (with some NH3/Ag(NH3)+), dilution by a factor of 10 actually decreases the amount of AgCl(s), not increase. I am in the process of debugging the simulation to see if I have made any mistakes, but this could be an epic fail on TSFX's account.