Actually it's 100% the opposite.

Look at your iron-carbon phase diagram: https://www.tf.uni-kiel.de/matwis/amat/iss/kap_9/illustr/fe_...

Cast iron contains a high concentration of carbon, which makes it brittle. To make steel you need a heavy oxidizer to extract the (reduced) carbon. This is usually done using a "basic oxygen furnace", which is basically injecting a high pressure, very high velocity stream of oxygen into the metal to game thermodynamics into combusting the carbon without combusting the iron.

This is correct, and the carbon is a byproduct. The reason is to get rid of the oxide in the iron-oxide compound.

The way I understand it, is that when mined they get iron-oxide, Fe(2)O(3), or Fe(3)O(4), so there is no carbon present.

They use the coke to make carbon monoxide (CO). And this process removes the oxide and adds carbon to iron to create pig iron.

From there it will be further processed to create different steels, alloys and iron products.

Wikipedia (https://en.wikipedia.org/wiki/Iron_ore) under the smelting paragraph describes the oxide removal process.

The coke that is mixed with the iron ore is the coke that heats the furnace. As the coke (carbon) reacts with oxygen, some of the carbon monoxide then strips oxygen from the iron oxide in the iron ore.

You sound like confused, from my limited knowledge, the making of steel is exactly opposite of what you said, they extract carbon atoms from iron, that makes carbon monoxide into carbon dioxide.