Could you have encountered an early adaptation of Gramophone of the Ages by Yefim Zozulya, published 1919?
Vandermeer's 2019 compendium, Big Book of Classic Fantasy claims to publish the first English translation (by Ekaterina Sedia) of the story but that's not to say it hasn't been adapted in other (Anglicised) forms before.
The rationale for the titular gramophone sounds really similar to your description:
...human voices, as well as other sounds, are recorded as invisible bumps on all inanimate objects near which they originated. Those bumps, according to Kuks's theory, are preserved for millenia, and new sounds merely deposit on top of the old ones, creating layers--just like dust, sand, and many minerals in nature.
The story is very grounded sci-fi. Kuks, the inventor of the gramophone, has been working on his theory for forty years before this breakthrough and wants to present his achievement to the Academy of Science. They don't listen to "well-known figures from history" but the revelations the gramophone unearths provides a social commentary; the story is set in a socialist Russia but the gramophone allows them to contrast their present utopia to a time "before socialism triumphed". In the story's climax, Kuks and his invention is rebuked for the history it reveals during its demonstration to the Academy.
Much less did they get the chance to listen to Michelangelo but in the course of testing the gramophone, Kuks, whose day job is as an advisor in the "Experience Workshop", suggests to a mason to take up the art of sculpture as he is more interested in shaping stones than breaking them apart.
This is a wild, wild, shot of course but while the scant details provided don't fit perfectly, it at least coincides.