As a (beginner) constructionist, the problem is not that the product "could be empty", but just that the product can't be constructed. I view this in a similar

As a (beginner) constructionist, the problem is not that the product "could be empty", but just that the product can't be constructed. I view this in a similar way to Russell's Paradox (while I'm not claiming the axiom of choice leads to a contradiction) -- we can define how we can construct new objects, and we can say you can't invoke the axiom of choice, but have to give an explicit choice function.