Little Details

No, pushing the arrow through seems reasonable if it's already through partially--meaning, it's already poking out the back. But say you've got an arrow in the meat of the thigh. It's not deep enough to hit the artery yet, but to avoid tearing the muscle you push it THROUGH the leg, and therefore THROUGH the artery: very bad! Much better to pull it out in that case!

The key thing is to know what organ/tissue/etc the arrow is already through, which often you can and sometimes you can't without exploring. Anatomy knowledge can tell you if you're in a safe spot or not, but there are so many variations! That said if the dang thing is already through, coming out the back of the person, yeah, probably ok to break off the fletching and push it though. Won't do any MORE damage, one hopes. Except that if it IS through something important, specifically a blood vessel, the shaft might be holding the hole it has made closed, and removing it will uncork that problem.

For example: an arrow going through the lateral thigh, front to back: I know that there is nothing there I can hurt other than muscle and maybe some small nerves. Pull it out if you like! Keep the wound clean and dressed; do NOT sew it up! But arrow through medial thigh, front to back: tricky! One or two major arteries in there, depending on the level, and a big honking vein. Is it already bleeding like crazy? If so, is the blood dark or bright? Is there a pulse in the foot? Those can help me guess if it's already hurt a vessel. If not bleeding very much, pulse normal in foot, and arrow all the way through, I would probably push it through, being ready to hold pressure if it starts spraying in my face, and open up the groin and the wound to find the blood vessel and repair it.

It's all context dependent, is what I am saying. There's no good rule--push it out VS pull it out VS leave it in place. It depends on what it might have hit! And depending on what organ it hit, there will be a totally different approach. For example, it hit the lung: 95% chance surgery is not needed for bleeding. BUT, a tube to drain the leaky air and what bleeding there is IS needed: a thoracostomy tube. Hit the heart: 100% surgery IS needed, but the heart is relatively trivial to fix or catastrophically NOT POSSIBLE to fix. A small penetrating would you can literally just use a skin stapler and be done with it; if a bunch of muscle is blown away, you're basically dead no matter what. BUT the pericardium must be opened, because even a small amount of blood in it can compress the heart and kill you.

What I am saying I guess is... uh, there's a reason it's taken me a decade to become a surgeon. If there were easy rules to follow here then my training would have been a lot shorter :)

Never fear, most people won't have any idea, so you can write what you like--and if you want some anatomical accuracy, I love explaining this shit and would love to help.