It'd also be an end to the project and would make the conjecture far less interesting.

IMO it would make the conjecture far more interesting, as it would be a surprise to most people who have thought about the problem.

Many natural questions would arise, starting with “Is this the only counterexample?”

Possibly, but it would join other false conjectures such as Euler's sum of powers conjecture - posed in 1769 and no counterexample found until 1966. There's only been three primitive counterexamples found so far.

(I got that from https://math.stackexchange.com/questions/514/conjectures-tha... which features some other false conjectures that may be of interest to you)

Not even the same implications. All empirical evidence strongly support the Goldbach conjecture. Any counterexample would mean an entire field of Mathematics has to be rewritten.