The sword was almost irretreviably part of Rhulad.
However I still find it strange that only Karsa was able to cut his arm off, and this leads me to my next question: would Rhulad's arm have grown back after that incident if they had left the situation as it was?
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You are implying that Rhulad was worthless and the sword was some kind of automated ascendantguided deathmachine. Learning through observation only capable of reproducing moves it has seen. I don't believe this to be true.
It was often quoted that Rhulad had little control over the sword especially in the beginning, Rhulad could not have perished if he had wanted to, the sword was on autopilot learning from each battle. There is a quote or two on this basis. Sure the exposure to battle increased Rhulad's prowess and his ability to become one with the sword resulting in a relatively smoother performance but in general I think the Sword is the real power behind the two of them.
Everyone was expecting Rhulad to have to die a few times before he conquered Karsa anyway, Karsa expectedly won the first battle but in the same instance he ended the war. The sword and Rhulad had no idea how to handle Karsa, although his tactics seemed fairly one-dimensional I'll give you that.