Don’t you need to get the bullet out before patching them up? I don’t remember ever seeing a movie where it’s implied that digging the bullet out is sufficient, only that it’s a necessary step.
I think that digging the bullet out before you can patch up the wound would result in losing a lost of blood? I'm not sure but at least if you get stabbed or have an arrow shot into you, then you shouldn't remove it before you are in a place where you can receive proper medical care.