Earlier quoted context omitted.
There are many cases where it's possible to prove that counterexamples must exist, without identifying a specific example. This kind of proof provides more insight into the problem than simply finding a counterexample.
Constructivists would surely disagree!
(Note that this is about excluded middle. There ARE constructive logics with interpretations of excluded middle, e.g. some forms of classical linear logic, but they do not play as nicely with other logics. Constructivists often reject even weak forms of choice for largely the same reasons--there are some forms of choice that are constructively valid in some logics, but these results often fail to hold true in more conventional logics. And the same is true for a whole host of related notions that proof assistants like Rocq reject by default, propositional extensionality (which says that two proofs of the same proposition are equal) and function extensionality (which says functions are equal whenever their results are equal on all the arguments in their domain -- which might seem obviously acceptable until you realize that it's false in most programming languages!) being prominent but much less discussed examples. It's all about remaining broadly compatible with lots of different types of reasoning, not because people think the reasoning is invalid per se).