An amputation wound would be massive, severing major veins and arterys, and likely done with dirty equipment. Why were they preferable to cleaning and bandaging the original wound?
I'm not an actual historian but I have a great interest in the Medieval Ages and realism for my DnD campaigns. I've touched upon this subject before, and hopefully the sources I have collected myself along with my recollections can serve as a foot-note until an actual historian can answer your question.
I'm going to challenge the assumption that amputations were considered safer. For example Ambroise Paré (c. 1510 – 20 December 1590, a French barber surgeon who served in that role for kings Henry II, Francis II, Charles IX and Henry III), says in his book Ambroise Paré IV: the early history of artificial limbs^(1) that patients with amputations were rare because most of them could not survive. Marit Van Cant (PhD researcher in Osteological Archeology at Brussels Heritage Lab) in her book - 9 Surviving Amputations: A Case of a Late-Medieval Femoral Amputation in the Rural Community of Moorsel (Belgium)^(2) - recounts the rarity of transfemoral amputations (that is, above the knee) and people surviving them, with her work mostly exploring the osteological evidence of a specific woman who seemingly survived one.
In an engraving in Vienna printed in Antwerp around 1560, a man is depicted with a bent-knee peg with four legs as a platform, meaning his leg is not amputated but instead leveraged as such (image example). Similarly, in another engraving from same place and time, a man is depicted with a bent-knee peg and a cane. Hieronymus Bosch, a dutch painter from Brabant (c. 1450 – 9 August 1516), similarly depicted people with many alternatives to amputation as you can view here. Yet again, here is a painting by Pieter Brugel the Elder (1525-69), showing people in a gathering some of which are on crutches or other similar devices.
In his book Cyrurgia^(3) (Surgery) Henri de Mondeville (c. 1260 – 1316) - a surgeon to two kings of France, Philip IV and Louis X, respectively - goes into detail of when, where and how to amputate and was allegedly the first surgeon to practice ligation of arteries (something that is documented in the chapter of amputation of limbs). In surgery, a ligature consists of a piece of thread (suture) tied around an anatomical structure, usually a blood vessel or another hollow structure to shut it off. Henri states that amputation occurs only when gangrene that has resisted all other treatments take place. Here, the book explains that to stop the rot to advance, you cut at the next joint; for example, if the rot takes root by the hand, amputate by the wrist, or if rot appears by the wrist, cut above the elbow. Interestingly enough, Henri claims that you cannot live an amputation above the elbow (i.e. if gangrene has spread to your upper arms) or above the knee (with gangrene to your thighs).
All of this is to say, the notion that amputations were preferred does not seem to be true; indeed, instead it seems amputations would serve as a last resort.
Again, I'm not a historian, and if mods feel this answer does not meet their standard - in particular the ability (or lack there-of) to engage with the original poster through a broader understanding of the subject matter with up to date, contemporary sources - then feel free to remove this comment. I thought it might give a glimpse into the spirit of the question asked; but again, a real historian might shed some light on this.
1: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3664166/
2: https://brill.com/view/book/edcoll/9789004363786/B9789004363786_010.xml
