Now, should the shooter have killed him? That is a whole another discussion.
The way you worded it, it could be a couple of different discussions at least.
The first point is, why are you even asking, do you think there was an option to stop him using deadly force where there was no risk of death??
The law recognizes that deadly force is exactly that, force that may result in death. The way our legal system works, intentionally killing someone is always a crime. Outside a courtroom many find it a distinction without a difference, but in court it matters, whether you intended to kill someone, or if they died as a result of being stopped by justifiable deadly force.
Next point, tied in with but slightly different from the first one, the justification for using deadly force on someone is to stop them from doing that to you. If you shoot them somewhere that death is unlikely, its possible that will not stop the attack. Also, there is a legal argument about only "shooting to wound" and that is, that since you only tried to wound them, then you did not actually believe deadly force was necessary.
AND, under our system if you do not truly believe deadly force is necessary, then your "defensive" use of deadly force can change to offensive action in the eyes of the law.
And this argument is also applied to "warning shots". If you do not believe that deadly force is absolutely necessary then you should not shoot.
I'm not a lawyer, and this is not valid legal advice, its just my observation of how things seem to work. (and my CYA)
And fwiw, asking "should the shooter have killed him" implies (to me) an execution /deliberate murder, like putting two rounds into the back of his head after he's already been stopped. That certainly isn't legal, and its probably not what you meant, but its what your phrasing brings to my mind.
In "gravest extreme" we shoot to STOP the threat, no more, no less. If the attacker dies as a result of being stopped, the law allows for that.