2 Fe2O3 + 3 C → 4 Fe + 3 CO2
Fe2O3 + 6 H2 → 2 Fe + 3 H2O
So switching to hydrogen could save about 500kg of CO2 per capita. Carbon capture of the fumes could help alleviate some 30% of that. Direct air capture is just not feasible, or wayy to expensive compared to just using H2.
In any case, even with H2; 321 kg of steel per capita will have to be reduced (main uses: construction, transport, industry, pipes, machines, weapons)
The problem is it costs a lot of energy, and so it ends up 4-5 times more expensive than oil from a well.
The real problem there though of course is that oil from the a well has always been cheating by not pricing in its externalities. To get an apples-to-apples comparison with synthetic fuel, fossil fuel would also have to be made at a minimum [0] carbon net neutral too, such as by running an atmospheric scrubber and ensuring that for every ton of carbon coming out of the ground a ton was getting captured again in an equivalently long term stable way.
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0: Fossil fuel extraction has always resulted in a ton of other externalities too, not just in terms of massive non-carbon pollution but geopolitical costs. That itself may well be a driver, if a country can switch fully to renewable/nuclear power and then run its entire economy solely off of that via grid/batteries/synthetic hydrocarbons the security implications alone are pretty massive. No more fossil fuel blackmail from hostile regimes, ever.
The basic mistake you're making here is thinking that "carbon free" is key. It is not, what matters is that a given energy source is carbon NEUTRAL. Anthropogenic global warming is driven primarily by net increase in atmospheric CO2 on our timescale. When CO2 in consumed from the atmosphere, and then shortly released again (textbook example being a plant fixing CO2/H2O into sugar and the getting eaten by an animal and the animal the metabolizing sugar back to CO2/H2O) the short term net change is zero. The problem has come from releasing stored carbon that was fixed in geologically long term ways, ie, fossil fuels.
We're used to thinking of "hydrocarbons" as equivalent to "fossil fuels" but that's not true. It's perfectly possible to directly synthesize any hydrocarbon from atmospheric CO2, water, and zero carbon energy like solar. Burning those hydrocarbons later will result in net zero change to atmospheric CO2, and thus are perfectly acceptable from an AGW POV (at ground level in particular burning fuels have other pollution issues that may be worth getting away from, but not global warming). The issue is "just" cost, it's much more inefficient and thus costlier to go solar -> fuel synthesis -> combustion -> useful work vs solar -> transmission -> battery -> work. But for applications where battery energy density is insufficient or other properties are required (like high performance aircraft that use fuel as coolant as well as energy) that could be worth it.
So green hydrogen directly faces not just batteries, but green methane, gasoline and other hydrocarbons. Which are far, far easier to work with and have many better properties than hydrogen itself. And on top of course feed seamlessly into existing infrastructure and system.