It’s important to realize that what a blacksmith does when forge welding iron (which is how iron was welded for a long time) is very different from modern welding. In forge welding, you hammer a convex shape in the two faces to be welded, then heat both and then hammer them together (with a flux in the middle).
This development is fairly obvious—you’re using heat to shape the material, and you want to stick two pieces of metal together… one leads to another.
However, forge-welding steel doesn’t work anywhere near as well as in wrought iron, so you need a different technique. Puddling the steel and then back-filling that puddle to create a strong joint is more or less the obvious method to connect the two together.
Notably, this later form of welding didn’t come about until the 19th century, when there was enough materials science & chemistry to come up with oxy-fuel welding and then various electric arc welding methods. I do imagine that many people went blind trying to figure out welding, but then you figure out you need a shield.