I never thought about it before but are the Iron Ingots we use in Minecraft pure Fe?
If its the same Iron we use in the real World then it has Carbon in it, so much that is becomes brittle.
Steel has less Carbon in it as Iron so is the IRL way to think about it abit confusing.
The coal used in steel production is in the blast furnace and is actually used to reduce the oxygen in the processed iron ore to make raw iron rather than steel. Basically you burn coal with to little oxygen to produce carbon monoxide which to some degree react with the oxygen (to make carbon dioxide) in the hematite (Fe2O3) which produces raw iron or pig iron with a carbon content of about 3-5%. The reason for the high carbon content is that the raw iron ore reacts with the coal during the refinement process. To make steel you mix the pig iron with oxygen so that some of that carbon is removed untill you have a low carbon steel which is usually 0.1-2% carbon.
So in essence if you have raw iron ingots (which has high carbon content due to the blast furnace) what you need to produce steel is oxygen, not coal.
What people miss out on in iron processing in minecraft is that you dont just smelt the iron ore with heat, you also need a reducing enviroment to remove the oxygen from the iron ore which is usually done with coal in a blast furnace.
You should require coal to produce pig iron ingots from iron ore (as a reducing agent, not necesarily heat, ie make coal or charcoal a requirement for producing iron)
To make steel you need to mix molten pig iron with oxygen to reduce the carbon content. (ie make oxygen a requirement to produce steel from iron)