There were other "right" things he could have done (called the police, hit the man over the head with a chair, etc.) that would have been less likely to be fatal
He needed to stop a deadly attack quickly.
If he'd called the police his mother would have died.
Hitting the guy over the head with a chair probably would not have done anything. If it had he would have had to defend himself against an enraged, murderous grown man or hit him hard enough to render him unconscious for a significant amount of time. That generally means a persistent coma. In other words, we're back to deadly force but deadly force.
In a situation like that keeping the innocent people alive is the only real concern. Doing it with as little harm to the attacker is a minor grace note to be played if you have the luxury. He didn't have the sheer physicality to easily and safely overpower the bad guy. He wasn't highly trained in arcane ways of harmlessly immobilizing people and didn't have access to specialized weaponry designed to do just that. He had to make a life or death decision within seconds in a chaotic environment. I wouldn't dream of second guessing the choice.
Grown men who are trying hard to kill with their bare hands is on a whole different level than what people see in the dojo. It's a far cry from what most ever go through including a lot of police officers. As usual I have to quote the great jurist Oliver Wendell Holmes "Calm reflection is not possible in the face of an upraised knife."