Steel is perfectly recyclable, but even then there is plenty of iron on earth. We won't be running out of iron.
Edit: Of course, steel is just a name of class of alloys - some of the steel types have a rarer elements like Mo, Ti, V...
Plus, steel is entirely recyclable. And it has some natural properties that make is relatively easy to recycle. It can be sorted with magnets, and it has a higher melting point than most impurities.
This way you get to reuse all the (enormous) existing infrastructure, as well.
We live on a thin crusty shell around a ball of iron.
Nevertheless, the mantle is made of a mixture of iron oxides, silicon dioxide and magnesium oxide, with small quantities of the other elements, so under the thin crust, even if there remain thousands of kilometers until the iron ball, there is nonetheless what is essentially a huge amount of iron ore.
... which is way beyond our reach.
On earth it replenishes too via meteorite strikes and by nuclear decay. I'm not sure was decays into iron, but I'm sure it's most things, given its name as the most stable element
Mind you, rust is quite similar to iron ore
Also I'm doubting "minimal processing" for steel. You have to dig up the ore with giant machines, transport huge amounts of it by train, smash it with a lot of energy and heavy equipment, melt it with a lot of energy and heavy equipment, etc., etc. This seems like the opposite of minimal?