Batman, and really most superheroes, have been massive cultural icons and popular figures for nearly a century. Probably every one of them is a vigilante by some definition. The entire reason why Batman can't kill is because he has no legal power and if he did he would absolutely be tried and charged for it.

Vigilantism is one of the most powerful instincts humans have, the desire for justice. The entire purpose of the legal system as a fucking concept is to satiate that desire enough by promising to take care of it for them. And the more they fail at the one single job they have, the more pronounced the call for it will become.