But the commenter has become emotionally involved with day-to-day sets and Cartesian products.

Not necessarily. At least, not necessarily more so than anyone becomes "emotionally involved" when deciding on axioms to use in a mathematical theory.

AC is after all independent of ZF. So of course no argument in favor of it can be constructed literally on the basis of the "original theory" (ZF). Saying that the Cartesian product of nonempty sets ought to be nonempty is an aesthetic statement about what the rules of the game should be, not a mistaken inference from the axioms of ZF.

Because of the independence, we logically have a "right to expect" anything we want (AC or not-AC). The choice is a matter of taste, and the tastes of the mathematical community evidently incline toward AC.

Note how different this is from your two previous examples regarding derivatives and Russell's paradox. Those involved outright logical errors; whereas in the case of AC, the commenter is making a legitimate aesthetic argument.