It's The Little Things

Izuku is All Might's biological son. Instead of choosing the successors of One For All, it is passed down from genetics. Inko dies during childbirth, leaving Izuku in the care of Yagi. Only problem is that he decides Izuku's worth by quirk and Izuku is a late bloomer. When he's six and One for All has yet to show itself, Yagi believes Izuku does not have it and abandons him in a shitty part of Japan. Abandoned and alone, Izuku is forced to survive on his own. Eventually, he becomes a vigilante and Eraser becomes Eraserdad. And I haven't even started on the cousin who prefers to stalk over communication to his beloved little cousin.