Astronomical dating agency
Better yet, one can determine within a very narrow tolerance where a given object was in the sky, and how long ago a spectacular coincidence event (such as an eclipse of the sun or moon) occurred.
If, then, a reliable historical record exists that mentions that event, then modern investigators can calculate when it occurred, with reference to whatever epoch one wishes to use, and with reference to modern historical records.
There is much discussion of accuracy and more of eclipses.
The only solid facts must surely come from verifying planet observation and then tracking those dates and comparing them to those given by the ancients. As you have found astronomical observations can indeed be used to help provide support for chronologies of the ancient world, no method is perfect however and there are problems with this.
The International Astronomical Union (IAU) was founded in 1919.
Its mission is to promote and safeguard the science of astronomy in all its aspects through international cooperation.
The end result is that for your example stone carving there might be a match with 560BCE, 552BCE, 544BCE, 536BCE or 528BCE.
Sometimes knowing that it must be one of those dates might be sufficient to combine with other data and pin down an exact match, and sometimes even being able to pin it down within 5 cycles or whatever the case may be might be superior to what was known before, but gaps and inaccuracies in ancient records make it difficult to pin down exact dates from historical astronomical observations.
Any record that sets a date is a good candidate for such synchronization.
All astronomical observations one can make related to the Solar system contain many layers of cycles.
Everything of course varies on the cycle of a year as the Earth orbits the Sun, however due to the motion of other bodies and slow periodic changes in Earth's orbit there are also longer period cycles.
Its individual members — structured in Divisions, Commissions, and Working groups — are professional astronomers from all over the world, at the Ph. level and beyond, and active in professional research and education in astronomy.
The IAU has 12557 Individual Members in 101 countries worldwide. In addition, the IAU collaborates with various scientific organizations all over the world.
A molecule once thought to be a useful marker for life as we know it has been discovered around a young star and at a comet for the first time, suggesting these ingredients are inherited during the planet-forming phase.