The “6,000 years” idea was probably not ancient people trying to date the physical age of planet Earth. It was a chronology built from the oldest civilizational memory available to them. For them, the beginning of “the world” meant the beginning of the human moral and social world they could narrate. Family, law, agriculture, violence, covenant, peoplehood, kingship, and civilization.
Were they wrong about the physical age of the Earth? Sure. But that does not mean they were lying, acting in bad faith, or randomly inventing numbers. They were still trying to be accurate, working with the records, memory, symbols, and genealogies available to them. You are not exposing a fraud. You are judging ancient memory as if it were modern geology.