As I understand it, it wasn't to work the carbon out of it, but instead to simply evenly spread the impurities throughout the whole sword, so there weren't any specific points of weakness
Obligatory "It wasn't folded 1000 times, it was folded 11 times giving it more than a thousand folds." Fun fact, the older traditional Japanese forges were basically assembled from mud found right there on the rivers where they found their low grade iron, and this process was kept alive by spiritualism and tradition conservatives, so it's honestly a miracle they assembled anything long and sharp at all. Most of the iron in Japan is considered low grade Pyrite at only 46% iron, while other places in the world had access to Magnetite with up to 72% iron before any refining.