Out of curiosity, what was it that made better metallurgy possible?

Gunpowder, which required Iron Working and Invention, and so on.

The industrial revolution has some very tightly coupled advances, The key advance was making iron with coal rather than using charcoal. This reduced the price, and a large increase in quantity of manufacture followed. One of the immediate triggers was that England was getting rather short of wood, and the use of coal as a substitute started for iron-making and heating.

The breakthrough in steelmaking was initially luck - some very low sulphur coal was found and used in steelmaking. But luck arises often out of greater quantities of usage, and perhaps that was the key here. It certainly wasn't science in the modern sense as the chemistry of what was going on wasn't really understood - certainly not by the practitioners of the time. Trial and error was therefore the key, and greater quantity of manufacture leads to more trials.