All Might tell Midoriya he can't be a hero when he's quirkless. Does that stop the green bean? Nope. Finding an abandoned toddler in an alleyway wasn't on his to do list though. But no one cared enough to care if he adopted the kid or not. And if anyone thought something small like a villainous quirk was enough to put Midoriya off, then they were very, very wrong. He knew what it was like, after all.