Just gonna ramble a bit 'cause I've got many thoughts and zero cohesion...
Why does Jin Kagemori work as a cleaner and take on all of the "dirty work" himself? Why is it that he is the only Kagemori brother who is no stranger to killing while neither of his older brothers have ever taken a life and for them to do so is treated as crossing a line?
Initially, I thought that he took on handling the Kagemori Clan's dirty work as a way of protecting his noticeably more sheltered older brothers - and I still think that is part of his motivation - but now that we know more about his background I think there is more to it.
I think the reason that Jin adopted the role of cleaner within the Clan has a lot to do with his own feelings of unworthiness of being a son and brother of the Kagemori family. Something about Jin's relationship with Gonzo stuck me as odd, especially when compared to his older brothers. I'm not just referring to the obvious example of Jin addressing Gonzo as "old man", while Hikaru uses the more familiar "dad" and Asuma the more respectful "father." I'm talking about how he comes across as obedient to the point of subservience to Gonzo. Wasn't it odd how during the meeting between the Kagemori inner circle and the Danno/Tadera family to discuss an alliance, Jin is the only brother who doesn't give an opinion and states he will just do as Gonzo says? Or how he doesn't protest Gonzo's decision on how to dispose of the gardener's body - unlike Hikaru - even though we later find out that he doesn't agree with it and ultimately goes around it? Then there's the way he gets miffed at Asuma for going off and doing his own thing instead of acting on the will of the "Kagemori Clan proper" aka Gonzo (okay to be fair though, Asuma did take a risk that could've set them back and he lost two people's lives in the process - oops). It's as though he feels that he cannot speak his mind within his own family and it's such a contrast to the other two Kagemori brothers who are comfortable enough in their positions within the family to express their own differing opinions, argue with, and even disobey their father.
With all this in mind, I couldn't help but get the impression that Jin was likely an illegitimate child, or at the very least he wasn't raised within the Kagemori Clan. And what do you know - turns out that's exactly what happened. His parents split up before he was born and he grew up having nothing to do with the Kagemori family until his heartless mother abandons him in the cold and leaves him with nothing more than the words "Go to the Kagemori Clan." The same family whose acts she despised so strongly and felt so disgusted by that she left while still pregnant. These are the people who she willing leaves her young son to. What does that say about how she views Jin and how he comes to see himself? The way she indifferently discards him is no better than how one would throw away a piece of trash. Imagine being discarded in such a manner by the person who should love you more than anyone. How could he possibly walk away from that without feeling a sense of worthlessness? Is it any surprise that the boy who was treated like trash later grows up to be the one who disposes of it?
I believe that despite their immediate acceptance of him, Jin still struggles with feeling more like a stray who showed up one day on the Kagemoris' door step than a true son of the Kagemori family. I think he feels that he needs to earn his keep by doing unpleasant jobs because he doesn't deserve the high position he has in the family by virtue of being Gonzo's son. This also explains his tendency to behave more as a yes-man to Gonzo than a son. It's as though he's always toeing a line and it's so heartbreaking.
I also believe that he chooses to uphold and take part in the abhorrent tasks necessary to protect the Clan as a way of getting back at his mother. I'm sure he's well aware of her disgust toward the dirty, heinous underbelly of the Kagemori Clan and takes revenge in his own way by continuing to support it. I don't really want to say too much about Gonzo's wives and how they impact his relationships with his sons since we just don't know enough them, but I get the feeling that part of the reason Hikaru and Asuma have little involvement with all the "disposal" within the Clan is because Gonzo knows that their mothers would not want have wanted them to become murderers. So, perhaps he has avoided putting his two oldest sons on that path out of respect for their mothers' wishes. As for his youngest son...well, there's no reason for him to have the same reservations.