I wonder if you could decarburize molten mild steel into wagon-wheel-tire-purity iron by blasting air or oxygen through it for a longer period of time than a steel mill would find profitable. This involves temperatures well above normal blacksmithing temperatures but it doesn't require a dangerous oxygen-free environment.

Interesting idea. So you're saying that the oxygen will bind more readily to the carbon than to the iron itself, and carry it off as COx? I'm not a chemist but I know iron loves to bind with oxygen (i.e. burn) at high temperatures; that's why oxyacetylene cutting torches work so well.

Yes, as I understand it, that's how steel mills make steels from pig iron, by blowing air through them to remove carbon (as well as things like sulfur and even phosphorus). It heats up the molten steel, too, so once you get it going you don't need to keep applying heat from the outside, so your furnace fire doesn't need to be able to reach the melting temperature of the final pure iron (1538°) but only the melting point of what you start with, say, cast iron (as low as 1147° but usually closer to 1400°). Have you done cast-iron casting?

I understand that when you're blacksmithing normally having your iron get hot enough to become a self-sustaining fire is a risk to be avoided rather than a goal but I feel like this might be less alarming if it's in a ladle covered up with a layer of flux and slag so that it stops burning whenever you stop blowing bubbles into it through a straw.

There's obviously the difficulty of what to make your straw out of. Apparently the standard answer in modern basic oxygen steelmaking is, "a water-cooled, copper tipped lance with 3–7 nozzles is lowered into it to within a few feet of the surface of the bath and high-purity oxygen at a pressure of 700–1,000 kilopascals (100–150 psi) is introduced at supersonic speed," but I imagine graphite or aluminum phosphate would work too and might be a bit less exciting.

Apparently the standard way of doing this from Han China until the invention of the Bessemer converter is a thing called "puddling" which sounds a bit less violent but also slower. Wikipedia explains, "Working as a two-man crew, a puddler and helper could produce about 1500 kg of iron in a 12-hour shift. The strenuous labour, heat and fumes caused puddlers to have a very short life expectancy, with most dying in their 30s."

So, maybe not something you want to do every day, but it seems like something you could probably do a few times in your lifetime—I'm guessing 1500 kg of iron would last you a good long time of hobby blacksmithing, and we have a lot more respiratory PPE now than we did in the 01800s.

I've always been curious about casting but never tried it. One of these days.

In some countries there are foundries and schools that do weekend classes where they walk you through the steps and already have all the necessary equipment and safety knowledge. You might see if you can sign up for one. Anything involving molten iron has a tendency to be, on the human scale, incredibly violent; as a consequence, it requires great delicacy and care.

Or you could start with pot metal or aluminum.