Premodern amputation (presumably also modern) was actually a very sophisticated medical procedure, even practiced under crude/battlefield tent conditions. I've actually discussed medieval amputation practices before, if you'd like to read in more depth.
Basically, the surgery involves making an incision down to the bone, peeling back the soft tissue, and sealing off the blood vessels. At that point, the bone is sawed through. The skin and other soft tissue that had been peeled back, are folded over the sawed-off end and sewn and bandaged shut.
One quick stroke with a leg-guillotine would also remove the soft tissue as well as not allow time to cauterize or tie off veins and arteries. Presumably a saw was also a bit more portable for emergency situations?
FYI, you'll find more discussion in this post