If it's not in self-defense, it's vigilantism. Not sure if anyone is dumb enough to think that killing someone we think is going to do something wrong is a defensible policy. Personally, I think people that wear MAGA hats are going to destroy our democracy with treasonous acts (which carry the death penalty, whereas child molestation does not), should I be allowed to murder them?
If someone was stalking my daughter, I would put a restraining order on that person! To take matters into my own hands, would be a different story, I would restrain myself from harming the individual. I don't know if that person would harm or kill me and then what would happen to my daughter after that!
Note how most of those who voted for the vigilante option are usually the same ones who believe all police departments across the country should be defunded, which in turn begs the question that if one were to believe that father was wrong and also the police are a problem and should not be called, what exactly would they consider to be a reasonable alternative then? Just sit back and become a victim, and just hope to survive the ordeal? That's pretty much the only other option, and it's neither a sane or rational one at that.
The phrase “he claims was stalking his daughter” jumped out at me right away. What if his claim is inaccurate, a misperception on his part? Best let the authorities deal with it, because even if he’s right, harassment of a minor, despicable as it is, is still not deserving of the death penalty.
I wouldn't use the words "public servant", but perhaps doing that as a form of self defense. In the particular case above, it appears the father did what he did out of genuine concern for his daughter whom he believed to be in danger, and pedophiles can and have murdered children after they were done abusing their victims, and that father didn't want to risk his daughter being the next such victim.