The underlying problem is that constructivism and non-constructive reasoning are using the word "exists" (and, relatedly, the logical disjunction) to mean very different things. The constructive meaning for "exists" is certainly more intuitive, so it makes sense that constructivists would want it by 'default'; but the non-constructive operator (which a constructivist would preferably understand as "is merely allowed to exist"), while somewhat more subtle, has a usefulness of its own.