(obligatory opening caveat - it's been several years since I last dug deeply into this; I might be behind and will happily defer to practicing medievalists in the sub.)

Battlefield forensics is an interesting if just-a-little-macabre subject; there've been a few medieval sites where a large number of remains from relatively major battles have been found, and they often end up being a kind of treasure trove for questions like yours.

One of the best known sites for that is the Battle of Visby (or sometimes Wisby), fought in Gotland in the summer of 1361. Mass graves at the site provide a huge number of bodies, around a thousand. It's a great find for people who are interested in a number of topics like battlefield injuries, osteology, the demographics of medieval armies, etc., because it's a huge sampling of presumably-representative casualties who all fell at the same time (27 July 1361 or maybe a day or two later for severe wounds) for the same reason (severe acquired stabbiness syndrome).

A significant chunk of the ones with wounds you could identify from skeletal evidence - not every penetrating injury strikes bone - had signs of perimortem wounds to their heads or legs, which are often (comparatively) more exposed than the rest of the body in a battle, especially when armor's involved. Those heads and legs were still attached however; you don't need to lop something off entirely before someone bleeds to death, obviously. I seem to recall references (which I'm trying without success to find right now; I might have the wrong battle) of a couple of people from the site who lost limbs, including one where it was implied both legs were lost in the same blow, but the way they were discussed suggests it was noteworthy because it was so unusual. The huge, huge majority would have died from less spectacular injuries like traumatic brain injury or simply bleeding to death.

Examination of the bodies at sites like Visby (and others - Towton's grave is much smaller but was also noted for the information gleaned off the bodies) actually suggests that quite a few of them were severely wounded - as in severely enough for marks to show on their skeletons - but managed to recover enough from their injuries to participate with a little less luck at another battle months or years later. As with the lethal injuries, this suggests taking some serious blows to limbs while still keeping use of them afterwards (albeit with spectacular scars).

Basically: it happened, sure, but on large medieval battlefields just about anything that can be done to a human body gets done to a human body. As with a lot of things in human experience the really memorable ones are often the outliers.

It's difficult to point to accessible decent sources on this one, since it's a subject that's practically designed to flood the internet with clickbaity articles and personal blog posts, and a lot of what's left is trapped behind paywalls. Some general discussion of it comes up in Larissa Tracy and Kelly DeVries' Wounds And Wound Repair in Medieval Culture which is available in part on Google Books here - Visby is discussed in some detail in Chapter 8. Robert Woonsam-Savage and (again) DeVries discuss battlefield injuries across numerous sites and referring to primary sources in the first chapter of Battle Trauma in Medieval Warfare, which you can read here.