To roughly repeat myself from a sibling comment: I may have been unclear. I didn't claim that the only way to disprove something in a constructive system is to produce a counterexample.

The part that I was referring to was the last statement from the OP: that "a proof of existence of a counterexample necessarily provides more insight than a counterexample". I can't imagine a constructivist would agree with that in general.

I'm the OP. In the cases where a proposition can be refuted by a proof that doesn't involve counterexamples, by its nature that proof will tell you something about the reason that the proposition is false.

Whereas a counterexample, on its own, proves the proposition false but doesn't necessarily tell you anything else.

The real difference in the constructive case is that there are fewer classes of proposition for which a proof without witnesses is possible.

(Edit: side note, I didn't explicitly say "necessarily" in my original comment. I suppose there could be exceptions, although I'm struggling to think of an example. Constructively speaking, the ball is in your court!)