Conversely, if your preferences are dependent, then there are p, A, B, C and D such that pA + (1-p)C > pB + (1-p)D and yet A ≤ B and C ≤ D. Then I can weakly money pump you, building on Eliezer's example. Assume you are in possession of pB + (1-p)D, with the first draw being to determine which of B or D you have. Then I can propose a binding contract: I will trade A for B and C for D after that first draw, replacing your current lottery with pA + (1-p)C. This is advantageous to you, so you will accept it. Then, after the first draw, I will propose to trade back B for A or D for C, another advantageous trade that you will accept. Congratulations! You have just been (weakly) money pumped.

This part is wrong, because you're assuming that after the draw to determine whether I have B or D, my preferences regarding A vs. B and C vs. D must be the same as they were before the draw. If I have A ≤ B and C ≤ D before the draw, but A > B and C > D after the draw, then you wouldn't be able to money pump me, even though my preferences violate Independence.

Why did you make this implicit assumption? I think it's because under Independence, your preferences can't change like that, without opening yourself to being time inconsistent and money pumped. But with non-Independent preferences, your preferences have to change as events occur in order to avoid time inconsistencies and being money pumped.