I think the bigger problem is the distribution of knowledge. Would French blacksmiths know of coal’s usefulness in the same century as it becomes widely used? Maybe in the printing press era, but it’s a bit of a stretch in the 13th century.
is an almost meaningless statement in the context of producing quality iron and steel.
"Charcoal" is a reasonably consistent product, "Coal" is an entire spectrum that ranges from [thermal coal] through to [metallurgical (coking) coal].
In a quality production pipeline there's a seperation between means of generating furnace heat (can be electric arc, can be thermal coal fired, can be charcoal) and a means of sourcing carbon for bonding with raw iron ores (coking coal).
In the context of low grade iron and poor steel manufacture the distinctions matter less.
Since England had lots of surface coal deposits they started exploiting them in the 13th century for heating. It got so bad that they had to ban coal use in London in 1306 because of the air pollution. Depending on the locality and trade networks, blacksmiths from this era could have had plenty of coal.
The shift in the 17th century was running out of the easy surface deposits and switching to subsurface mining with chain pumps (driven by water wheels at first). That followed from centuries of development and resource exploitation.