Your analogy might apply for internal, subjective qualities that we historically associated closely (if not exclusively) with verbal capability -- like sentience or self-awareness or salience or moral reasoning -- but pain is something more like smell or taste.

Most wouldn't readily accept that a Claude instance or a mysterious teletype smells lavendar just because it output a text that claimed to, even if it characterized the smell of lavendar through beautiful and relatable prose. Only if we had some independent reason to believe that it was capable of smell would we consider those words to be true testimony to the experience.

Pain, as widely used in matters of ethics, is in that class. We traditionally recognize it in some living bodies (only a rare few capable of language) and accept that, as living bodies, humans who communicate about it with seeming sincerity are very likely to be experiencing it. Although even then, we may remain suspect if their words are our only source of knowledge or if we have some deeper intuition about why those might not reflect a real happening.