The time is coming when he will start to target evildoers within the city; as Gluttony gets greater, he'll have no choice about this.
He's already recognized that he's a monster. Greed has made it very clear to him what will occur if he doesn't keep his Gluttony skill fed, that he would go on a rampage without regard for who he kills.
The surrounding monsters just aren't going to be enough.
Combined with his desire for revenge upon those who mistreated him...
On another, related, note. It's clear that Knight Heart cares for him as an individual. I suspect that she will be the Romance focus for Our Hero.
It is inevitable that she will learn of his skill, and what it requires of him to keep it under control. The conflict this will cause between her personal feelings for him and her abhorrence of his vigilante activities is something the Author can't let slide.
When he does start targeting those inside the city who are monsters in human form, which clearly will include the family he's already aware of, it would be good if he can somehow document their crimes, such that when she finds out just what he is doing, he will be able to show her that he has been targeting those deserving of death whom the justice system would not be able to touch.