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Stuart_Armstrong comments on Median utility rather than mean? - Less Wrong Discussion
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But notice you had two decision points there.
Intransitivity breaks your decision system with a single decision point; dependence does not. Hence a single policy decision has to be transitive, but need not be independent.
The first decision is immediately canceled and has no effect on your utility, hence it isn't really a relevant decision point.
More generally, the independence axiom makes sure that the outcome of your decision process is not affected by bad options that are available to you.
Except that median-maximising respects independence for options that are available to you (or can be trivially tweaked to do so). It only violates independence for hypothetical bad options that will never be available to you.