Broken bones can be one lasting complication of damage in a game - for such game systems that support it. Many games don't have any mechanics for broken bones, the venerable Dungeons & Dragons among them, and simply abstract damage as a number subtracted from a pool of some sort of points representing a character's health.
Other games have wound systems which have varying levels of explicitness in assigning damage to semi-abstract "wounds" in specific body parts, all the way up to specific injuries such as broken bones, torn ligaments, lacerated tendons, and so on. As usual, adding more details means more book-keeping work and effort in tracking things, which may take time away form game play. So there's a balance to be struck, and where you put that balance depends on player preferences.
Try a few different game systems and see what level of detail you like.
Commentary by memnarch (who has not seen the movie)
How lucky! Yanni managed to find the cockpit very easily this time! How unlucky! Yanni managed to easily find the cockpit! Chewie probably knows quite well that Yanni doesn't really know how any of the controls work, so there's still a chance to prevent any intentional sabotage. Sure, cockpit controls might be relatively standardized, but with how simple the Falcon's floor layout is and the confusion it still causes, I bet that Yanni has no clue what any of the switches do. If the party is lucky, none of them are a self-destruct switch. If the party is really lucky, Rey will make it to the cockpit before anything bad happens and be able to keep the Falcon flying.
Callback! Seeing callbacks to the original movies like this are quite fun. I, too, would also think it not wise to set a Wookiee for fear of upsetting them more. I've been fortunate enough to not sprain or break any bones, but I have heard bone setting can be quite painful, especially when not done properly. Wookiee anatomy might be just different enough than human (beyond the heavy fur coat) that one would need to be very careful to adjust from the usual methods. Finn may or may not be able to fix the arm without better medical tools and Wookiee safe painkillers, but at least he shouldn't need to club Chewbacca unconscious first! That alone should give Finn a good chance of success, if Chewie can cooperate.
Commentary by Keybounce (who has not seen the movie)
"It's not wise to set up a Wookiee". That's gotta be the next version of this line at some point. Now, how does shouting hurt an injured arm?
When GMs start passing out notes, players start to worry because it means information is not being shared and people won't know what to do. The only thing more frightening is when players start passing notes back. And if everyone else is passing notes, you had better pass a note yourself, right?
I once passed a note that said, "I'm not necessarily doing anything, but since everyone else was passing you something I figured I'd join in." In fairness, not only was it a Paranoia game, but Friend Computer sent back a note thanking me for participating in things.
GM: Yanni, you enter the Falcon’s cockpit.
GM: Jim, take this note. And Ben, one for you.
Yanni: I start flipping switches!
Chewbacca: Do not let him do that!
GM: The strain of shouting makes your injured arm throb.
Finn: Do human painkillers work on Wookiees?
Chewbacca: Let me kill you and we’ll see how my pain is! Argh!! I think that blast broke a bone!
Finn: I’m a trained field medic. I can fix that bone for you.
Chewbacca: Be careful!
Chewbacca: It’s not wise to set a Wookiee.