
Tell me this, would you be comfortable with anyone having right over people’s lives as executioner with no oversight whatsoever. Tell me this, would you be okay with cops having the legal right to execute criminals on the street? No, obviously not, the issue here isn’t necessarily the pragmatic outcome of killing bad people, it’s that no one should have that power. The idea of killing those who would do harm to others is nice on the surface, but then it leads to the next question, can we trust anyone to wield that power of other people.

If a situation deemed it as necessary then yes. at the very least people should know it was necessary like how hawks killed twice. Even if it garners infamy, a hero should act accordingly which makes the best outcome. Let's say he got captured, who's to say that would be the end of it. To his predecessors he's someone too optimistic since his comrades never experienced being surrounded by death. Hesitance to act may just create more destruction. Remember that all might cause of hesitation failed to defeat all for one back then, resulting in more noumus created, lost of other's quirk and deaths while in hiding. Of course plot armor would make him able to save everyone unlike his predecessors who had their comrades dead.

Look, It's the same situation when there's a hostage situation. Police will be forced to decide if they will shoot in order to save the person. Hawks killed Twice because he knows what would happen if he let him be. Punishment or not killing him would stabilize the situation more easily. I'm not saying he should used it to every villain. I'm saying he should act that would help people. It's just like naruto all over again if he tried to save him.

Read "Trash of the count's family" and "The hero returns" (they're kr webnovels). They're both heroes but reject the idea of being called heroes. They will kill if necessary. Especially "the hero returns" he is righteous but can still kill people.
Deku's heroism is only optimism at best. He had not experienced people's death nor will ever be due to plot armor. That's why it's annoying. His hardships are atmost bullying, training and broken bones.
I hate how deku doesn't want to kill. The disaster wouldn't be big in the first place if there was death sentence to all for one. Him trying to save shigaraki who probably killed dozens or hundred of people (directly or indirectly) is like an insult to his predecessors who had countless comrades sacrificed. even if he was being manipulated now, it doesn't change his sins. Deku's idea of justice is inflexible, as long as they look like they're asking for help he would save them even if the guy committed massacres.