I think maybe a better way of explaining it would be that an uninformative proof by definition needs to be based on proving that the set under consideration must be inhabited without ever defining an object in that set. This generally means you must show the set is inhabited by exploring some abstract properties of the set itself. A single counterexample, by contrast, by itself is a direct proof that the set is inhabited, so you don't necessarily learn any other interesting properties about the set. So it's not really about constructive vs. non-constructive, I think it's closer to e.g. the idea that point-free stuff tends to be more beautiful and meaningful than pointed stuff (which I think most mathematicians would agree with and which really has nothing to do with intuitionism per se).

In this case, I think part of the problem is that there was kind of no good reason to think the Jacobian conjecture was true in > 2 dimensions other than it being kind of hard to find counterexamples. So a really interesting disproof would be one that, e.g., was able to exhaustively classify the counterexamples, or showed why it seemed in practice to be hard to come up with functions violating the conjecture. AFAIK, this doesn't really accomplish either of those things, not even after you learn the procedure that constructed the function -- it kind of tells you why we should have expected to find a counterexample but not how rare such counterexamples are.