Determination of carbon content of steel by liquid chemistry
I think cuppellate is a foreign word meaning "burn" essentially. 

The basic technique would be to have the iron sample in a small ceramic boat which is put into a hot stream of $\ce{O2}$ so the iron and carbon both oxidize. The iron oxide stays in the boat, and the $\ce{CO2}$ is sweep away and trapped/reacted somehow. We talking small samples 10 grams perhaps. 

If you try to dissolve the steel in an acid solution then the $\ce{CO2}$ gas is formed in solution. It is really hard to get a small amount of $\ce{CO2}$ out of solution given that you are starting with a small amount to begin with. Also if you try to scale up such a reaction it gets to be a problem. A $2~\mathrm{cm^3}$ block of iron takes a long time to dissolve in acid. 

So choosing the analytical method depends on a lot of factors. Time per sample, sensitivity, and overall cost being the main ones.