In general, reciprocity works well across a broad range of international relations, (visas, tariffs, mutual defence/non-aggression/..., and otherwise).

As an idea, it's not "too dumb to believe."

Indeed, it's one of the places the US' draconian visa policies often come back to bite it. If you're travelling to any developing country with reciprocity on a US passport, you can expect to be treated horribly.