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.
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.
Thus I will relax my criticism of using iron. If that was available nearby of course it would be used. Though I'm only relaxing as I still suspect they are using late 1800's refining to get the amount of steel needed for the chisels. (the hammers could still be pure iron - we don't see heat treatment used on them, and in any case hitting hardened steel on hardened steel will cause something to break). (case hardening would have been known in the 13th century and result in the needed steel, but that heat treats differently so I'm not sure what would have been used).
However even if good ore is found nearby, refining it takes a lot of energy (read wood) and so I would expect they avoid using iron just because of the amount of additional labor to make it.
Speaking of wood. Their blacksmith appears to be using coal not charcoal. This isn't clear, but they probably should be making the charcoal for the blacksmith onsite not using coal. Coal wasn't very important until 17th century England.
While I'm at it, I just noticed their great tower is round. I'm not sure if that would be correct. A cannon was becoming advanced enough to need round towers in the 13th century, but I'm not sure of the timelines, and the thickness of the walls matter. The Chinese built thicker walled castle and so they didn't see gunpoweder/canons as useful tools, but European castle walls were often (not always!) thin enough that a cannon could destroy them. Thicker walls would have then been used for castles in response, and also round towers so that the stones beside could help take the impact of a cannonball that otherwise would go through.
Speaking of stone. Odds are the small local lord would have built their castle out of wood not stone. Wood would have been much less likely to survive to today so we see ruins of stone castles while the wood burned or rotted away.