Could the clues to ancient alien civilizations be hiding in moon dust?

Most of our experience with the Search for Extraterrestrial Intelligence (SETI) has focused on capturing radio signals that alien species have sent out, whether intentionally or unintentionally. That creates a huge "synchronicity" problem—what if there aren't any alien civilizations broadcasting radio signals now, but there were a billion years ago? The Milky Way is around 13 billion years old—hoping that we exist at the same time as an alien civilization that happens to be actively messaging is a huge leap of faith.