As far as I understand - yes, I could look deeper into it, admittedly what I know is superficial, I've not made a pointed study of it at all - these two elements and their interplay among many elements including fire, seem significant for at least many Japanese tool smiths using pine charcoal fired forging and also the most highly regarded smith of Sweden who once preferred English coal but in the end made due with Polish coal. In these two examples it is the carbon that is determinate in forging as well as the heat treatment. It is the carbon that largely defines the cutting ability which the steel has.
It seems you are telling that temperature and temperature alone is the concern. In my ignorance, I cannot believe it.
E.DB.