Limb amputations don't work the way you think. In fact, there is a modern procedure doing basically what you're advocating called a guillotine amputation. I am not linking to any images here, but if have a reasonably strong stomach, I suggest you Google image search it, and you'll have your answer. A guillotine is a salvage operation -- it is for when you need to remove an overwhelming infection FAST (think necrotizing fasciitis or gas gangrene, what the media calls a "flesh eating infection"). You then leave the wound open and perform an actual amputation later that will allow you to close.

And that's the problem with a guillotine amputation -- you can't CLOSE either the wound, or the vessels that will soon be gushing blood. You can't make precision cuts of disarticulate bones (think about carving a turkey with either a guillotine versus your traditional carving knife). Military surgeons have been amputating limbs for presumably as long as people have been crushing, stabbing, or slicing them with weapons. But since the time of Ambroise Pare (16th century), amputation had become fairly savvy, with suturing shut both blood vessels and skin (as opposed to cauterizing). I'm actually fairly deficient in my surgical history, especially pre- the 19th century, but my understanding is that the fundamental techniques are still followed in many types of amputations today.

So -- a guillotine WOULD be faster and less painful, and would probably kill just about everyone it was used on, unless another amputation was performed later.