The Punisher is the very definition of an anti-hero. The last name on his list is his own. He understands that he literally shouldn't exist, but has to because society has, at the upper echelons, become so corrupt that only an immortal and nearly invulnerable being could possibly try to take them down.

Regardless, Frank Castle views himself as part of the problem, and will absolutely execute himself as soon as there are no more corrupt rich people or politicians in his world.

Edit: I know basically nothing about Moon Knight. I kinda look at him as a Deadpool type character. Insignificant enough that one can completely ignore him, but he just keeps popping up for some reason, mostly because he can swear from what I can tell, and yes, this applies to both Moon Kinght, and Deadpool.