Is there any good, one-paragraph explanation of the principle and importance of the counterexample in logic by a modern logician (19th to 21st century)?
See Evert Willem Beth's Aspect of Modern Logic (Springer, original ed.1967), page 10:
The argument [above] has a particular property: both premisses are true, but the conclusion is false.
We say that we, by a substitution of the terms [...] obtain a counterexample to judge the soundness [read: validity] of the argument. Making use of the concept counterexample, we can now formulate the fundamental criterion for the soundness [validity] of arguments as follows:
An argument is sound [valid] if and only if it admits no counterexample.
This criterion was known already to Aristotle, and it has been applied as long as mankind has tried to reason logically. Its fundamental character, however, has been understood only much later. In 1955 [E.W.Beth, 'Semantic Entailment and Formal Derivability' (1955)] I myself have shown how one can construct logic in a very simple and transparent way directly on the basis of the fundamental criterion.
Here Beth is alluding to the Semantic tableaux method, invented by him and simplified by Jaakko Hintikka and Raymond Smullyan. The method was originally named: "no-counterexample" proof procedure.