Isn't that wrought iron production? Too low of carbon content because the iron doesn't melt? We started getting cast iron once we could get the temperature up

Isn't that wrought iron production? Too low of carbon content because the iron doesn't melt? We started getting cast iron once we could get the temperature up to melting, then the optimized steels once we figured out how to add oxygen to the cast iron and burn out the excess carbon. It's still possible to get higher carbon content out of a bloomery to get steel, but highly unlikely... And you'd get what, a nail's worth from 3 people's labour over a couple of days? {Mine the ore/grab it from a swamp, bake it, make charcoal, make the bloomery, run the bloomery, consolidate the bloom}