This is not the issue. When expected value is well-defined, it is perfectly fair to compute E[X] = E[E[X | A]], where E[X | A] is a function of random variable A ("conditional expectation"). If E[X | A] > A for every particular value of A, then E[X] > E[A], whenever the expected values E[X], E[A], and E[X - A] are all well-defined.

The issue in this case is that E[X - A] is not well-defined. The expected profit from switching is given by an infinite series whose positive and negative components are each of infinite total magnitude, thus yielding different sums when bracketed differently ("conditional convergence"; just a coincidence that the word "conditional" comes up here again).