I've only read the MHA novel and the short story comics, but I've seen everything in the anime up to season five. The one thing I did not understand was Overhaul's need for violence. Like maybe his beautiful eyes distracted me but, from what I saw: he was an orphan who was taken in by the Yakuza. Then became their doctor, and when Eri's quirk manifested that's when he went off the deep end and started experimenting on her. Put her paw paw in a coma and just became a bad news bear for everybody, and for what? ( Please tell me.)
Again maybe his glistening orbs distracted me, but that's been keeping me awake at night.
(Love the pfp, by the way)
Overhaul, lets just call him Kai, is the young new leader of the Shie Hassaikai. He replaced the previous leader by using his quirk on him. The reason why is because he hates quirks. Why? Because he views it as some sort of disease which he isn't wrong, quirks are not normal and weren't normal till the larger population gained them. He doesn't necessaryly like violence its more control. He believes that he could torture Eri to get some of her dna and than make a drug with it that erases quirks.
Quirks are an illness, no im not playing advocate that is what they are. Quirks are considered illness that are unnatural but people have become so used to them and on top of that they benefit from them. Kai believed that is he could rid the world of quirks he could make it a better place, that the world would be better and the yakuza would return to its former glory.
His other and more believable reason for hating quirks is the yakuza. They are monitored and have no ground for their things. They practically dont exist and thats all thanks to quirks. Quirks brought a rise to heroes and with heroes brought organize crime down on its knees. No one cared to join or could even profit off of yakuza extortion dealings, many yakuza groups disbanded after a certain amount of time because they coudlnt really do crime. This is what had Kai go this far. Remember that he said he was doing this for his old man and for the yakuza. He wasn't disillusioned by the world of quirks and had a good point in disliking them yet the execution of his beliefs lead to the torture of an innocent girl he was suppose to father.
I hope this answered your question. Feel free to send in more.