The reason for Archimedes's excitement is that he had discovered a way to determine if a gold smith or jeweler had debased the gold used in making crowns, rings and coins. Based on Archimedes' excitement, we can assume debasing gold was an ancient and pervasive problem.
Like many ancient problems, there were other ancient solutions. Archimedes's solution was unique because the item did not need to be melted down. Melting down crowns and coins is problematic because even if you determine the gold is pure the next crown-maker or jeweler might debase the metal and you still don't know.
The ancient solution was to melt the golden object and to sprinkle saltpeter or nitre on the bead or puddle of liquid metal. Saltpeter is potassium nitrate (nitre is sodium nitrate) and both are strong oxidizers...Saltpeter is the primary component of black powder for that reason.
The saltpeter melts and covers the molten bead. If there are any metals other than gold in the golden object, the saltpeter will cause an opaque film of copper or silver or tin or zinc oxides to form on the surface of the bead, almost like a cataract over an eye.
Purifying debased gold involves multiple sprinkles of saltpeter followed by multiple sprinkles of flux (borax + lime+ lye + fine sand) and skimming off the resulting skin of slag. ---Note: A cullet of ground glass makes an exceptionally fine flux but was not readily available to the ancients---
The refiner knows he is done when he can sprinkle saltpeter over the molten bead and it does not skin over. That is, when he can see his reflection in the molten bead exactly the same way a mother can look into her baby's eyes and see her reflection when she kisses his face.
Like all things fire, the process of purifying and proving gold would have been of great interest to young boys and they would all have become intimately familiar with the process if they grew up in a town large enough to have somebody who knew how to assay gold.
One of the readings at my aunt's funeral was Wisdom 3:1-9 part of which reads:
As gold in the furnace, he proved them, and as sacrificial offerings he took them to himself.
The priest's homily briefly described the process of purifying gold and then he hammered:
"The goldsmith KNOWS the gold is pure and that he is done refining when he can look at the bead of gold and see his own face. When ancient peoples heard the phrase "gold that is proven in fire" the obvious imagery was that God looked down on the deceased's soul and clearly saw the reflection his own face. That is when God claims his own and takes them up to heaven."
The second thing that would have been immediately obvious to the ancient listener or reader is that gold purification is an iterative process. It is never a one-shot deal. Thus the endless trials and tribulations of life both purify and prove the believer.