Where did this guy learn Martial arts? line up your wrist and your posture! You can absolutely keep punching with a broken hand. Boxers do it all the time. Golota was a great example of carrying on the fight with a broken hand.
Why are you punching someone when you have a gun and a knife?
not a fan of his movies nor an Aikido practitioner, but If you pay attention to his arms and positioning you can see that it might look like his hands are always lowered but he raise and extends his hands to react to the incoming force. They are lowered again when he is in control of the situation. This gives you the appearance of always having his hands down. And attacks when he controls the opponents' balance. You don't always need you hands around your head. You lower them when out of range.