When you are amputating a body part and that action gets interrupted your progress should be saved has it makes no sense that you have to cut that body part all over again.
True, to add to both this and Overload's suggestion, the whole amputation process could be based on limb damage. It could cause limb damage throughout the cutting process, and instead of it "saving the progress" in case of an interruption, you are allowed to instead swing the cutting item faster during the amputation based on current limb damage, therefore preserving the whole aspect of the amputation minigame, technically saving progress, and also making it easier to amputate already severely damaged limbs.