why doesn't we count earth's inner rings to see how old our planet is?

We can't find a saw big enough to cut it in half.

They conducted this experiment about 6,000 years ago. The work that was published on the subject was so incredibly authoritative that they didn't even have to list their methods or resources, and no one has questioned it in a long, long time. They found that the Earth was a couple hundred years old ~6,000 years ago, so the Earth is currently ~6,000 years old.