Do clothes washed in automatic washing machines become pure, despite being washed with impure clothes, due to the repeated drawing and draining of water?

What you have mentioned is correct. If the impurity is hukmiyyah (non-substantive), then it is sufficient for water to flow over it to purify it, and the flow of new water over the impurity is enough to purify it. However, if the impurity is ayniyyah (substantive), then it must be washed until it is removed, and the water that separates from it before its removal is impure.