Adam,
you are not correct in your assumptions.
Stefan Mäder has done some interesting research on early medieval blades (Merovingian) and found that they were made very closely to the Japanese way. They had a tough blade body with a steel cutting edge, and they were heat-treated with differentially hardening technique.
Quality differences in early steel making were caused by the process temperatures. Higher temperatures above 1.350°C in the early blast-furnaces up to melting temperature (1.538°C) resulted in a much more effective output of cast iron while the TATARA (as well as the European bloomery kilns) had only a max. degree of efficiency in the range of 30% or a tiny bit more. On the other hand, TAMAHAGANE (as well as bloomery iron and steel) is very pure in alloying metals and - besides carbon - does not contain many impurities after being processed in the forge.
Cast iron, on the other side, cannot be forged because of its very high carbon content. It has to undergo different methods to reduce carbon to be malleable. When the Europeans improved their methods using industrial dimensions, they were able to increase their production so they could export their steel.
All this has to do with the desired steel quality. While Japan could produce the best (finest/purest) steel of their era, Europe had a much bigger output of iron and steel of different qualities, ranging from very good blade-steel to minor iron for horseshoes or wagon-wheel tires.