It was just bad luck really. It wasn't like Izuku was looking for a robbery in progress, or called Endeavor to handle it, or asked to be quirkless. But all these facts combined resulted in Endeavor deciding that Izuku was part of the robbery and every cop and judge that Izuku saw not listening to a word he said after they saw the bright read "Quirkless" stamped in big bold letters on his file.
It was also just bad luck that Izuku had just turned 10, meaning that, in the eyes of the law for the quirkless, he was an adult. But when had Izuku ever had good luck? His mom was killed a few years back when his apartment building caught on fire due to a wild shot at a criminal by the flame hero himself, not that he ever admitted to it or helped any of the people he had just made homeless, that would cause Endeavor to drop in the hero rankings and he couldn't let that happen.
So, Izuku Midoriya was homeless and alone for almost three years. He still went to school of course. he went there to be yelled at be his peers and teachers, to have his homework and lunch stolen, and to be burned by his so-called friend "Kacchan" every day. Every day Izuku spent his school day running from bullies and his nights working hiding from adults. Every so often a nice adult would give Izuku a few hours work for a small wage, but it wasn't very often as he wasn't able to do a whole lot, and that money would always go to replacing his ruined school supplies and uniforms so there wasn't much left for food.
It was on the streets that Izuku learned to fight. He learned to fight dirty, to strike fast and hard, and to use his small stature to throw off his opponents. Izuku soon earned a reputation as "the feral child of the undercity" where he lived. People said he fought like an animal, grabbing onto a target before striking all their weakest points with his fist, feet, elbows, knees, even teeth, throwing them to the ground and getting away before they could even think about getting up. People said he could fight fast and analyze faster. A few gangs had tried to recruit Izuku, but they always left with new bruises. So, instead of being taken care of as most children are at that age, Izuku took care of everyone else.
Izuku walked the women of the night to and from the brothel where they worked, he helped guide lost tourists out of the undercity, he even kept all cops and all heroes away from the somewhat honest working citizens of the undercity. But he couldn't keep out all the villains.
Izuku managed to keep out most of the more violent villains, not liking any that dared to hurt any of the hard-working citizens of the undercity. However, there were a few that Izuku was forced to tolerate as they brought business and sometimes ran a few of the honest businesses in the area. Izuku couldn't put honest people out of a job just because they happened to be working for a villain! They had families to feed after all.
So, the feral child of the undercity allowed some of the nicer villains to take refuge in the undercity. The only one there without Izuku's approval was someone the people called "the boogieman of the underground". Izuku had never seen them, but he knew they resided in the nicer area of the under city. This boogieman seemed to have a hand in every business in the area, every drug deal, and every gang, but he never caused chaos.
The green haired boy followed the Boogieman's work closely. They seemed meticulous, never letting innocents get hurt, nor causing panic that could attract heroes. As far a Izuku was concerned, this boogieman of the underground was better than almost every hero that earned a silence.
So, the Boogieman of the underground gained the support of the feral child of the undercity. Sometimes his henchmen would return to tell him about a blur of green that helped them in their mission. After a while, these assists became common, and people stopped questioning them. Everyone but all for one.
When all for one heard about a child digging into his plans, he thought he would have to ether recruit them or kill them. However, as far as the child dug, they never did anything. There were no police reports files (all for one would know as he had a few moles in every police department) nor were there any attempts to stop him.
It was almost as if this kid simply wanted to know what All for one was up to. All for one decided to look back into this kid the way they had looked into him. Imagine the surprise All for one felt when he spoke to a concerned prostitute to discover that the so-called "feral child" that had been adopted by the undercity had approved of him to be in his city and has gone as far as to assist his men when they needed help on a job.
Intrigued, All for one had some associates dig up the boys past. All for one was baffled to say the least to find that a boy who had had his home and family stolen from him by a careless hero who wouldn't even admit to any wrongdoing, would go out of his way to guard a lower income area such as the undercity. Especially with the way the school system treated him for his quirklessness.
All for one resolved to recruit the boy when he got older, perhaps after the world ridded him of his naivety. Such an analytical mind with such a fighting spirit could be very beneficial to him in the future. As long as the boy doesn't get himself killed before then.
All for one wanted a way to keep the boy safe without him knowing or losing his edge. The perfect opportunity soon presented itself with the boy's proximity to a robbery and will one phone call, the most prejudice hero could be called in and would no doubt assume the feral child guilty with no evidence.
All for one knew how the justice system treated the quirkless. The boy would be thrown in the farthest hole the judge assigned to him could find. And will adults no less! What better way to keep the kid sharp than to throw him in jail? How much did some lawmakers hate the quirkless to make consider them legal adults at age ten?
In a few short years, All for one would go back for the feral child of the undercity, but until then, All for one simply relished in his good luck.