So first thing first who said anything about only killing white people. That is you imagining something my post/idea isn't trying to be. It isn't about trying to be "better". The concept of Punisher is about what I call a "serial killer" anybody picking up and doing what he does regardless of targets would have some of the same issues. No where I am saying this is to create a better punisher conceptwise Punisher is a sort of revenge power fantasy about a person who always gets the criminal regardless of the law. Punisher is a flawed concept this isn't about getting away from the flawed concept.
Now with that said there is a danger in pretending the "otherside" is just as bad. So no I am not afraid of empowering some rogue vigilante person who thinks killing white supremacist or corrupt cops is okay BUT if we are talking about one having to happen by nature of character being around influencing the mentality of cops and soldiers to be more extreme and some random person to be more extreme I know which one is worse.
But again this isn't something that serious fiction really doesn't have that much power this just trying to get a culture change about The Punisher and move Punisher to the same status in fiction as say Hannibal Lector or Joker. It is just about making the character thus symbol not appealing to certain groups that they would public wear the symbol. There is a reason why we aren't having this convo about the Joker but that said when the movie was released there some similar convos about the glorification of the Joker by some people. My suggestion of another character for Punisher was only to speed along the process of a culture change not make a better punisher other than the fact it would be discouraging military, police, and militia not to wear the symbol.