AN: So, for this, All Might didn't drop the sludge villain when he leapt away with Izuku on his leg and All for One is actually Izuku's dad. This idea involves though, what happens when All for One learns of that meeting with All Might and still has his protective father instincts. More info at the bottom.
One Man's Trash
Swaying slightly, Izuku staggered into his home, tears still dripping from his empty eyes as he recalled that last image of the skeletal All Might, thanking him for keeping his injury a secret, telling him to give up on being a hero, and departing with the captured sludge villain in hand. "Izuku!" his mother's voice called out in relief as he opened the door, "I was getting…worried. Oh no baby, what happened?!" his mother rushed forward, embracing him as she saw his disheveled state, his tears, and his broken expression.
"I…I met All Might." He got out, fists shaking as the feelings of pain, betrayal, and bitterness surged and boiled in his gut, "I asked…I asked him if I could be a hero. He said to give up."
"He…what?!" a new voice roared in fury as Izuku turned to his mother's phone where it lay on speaker, recognizing his father's voice from the other end.
"There was a sludge villain." Izuku admitted, his tone dead as his crushed dreams "I got knocked out by All Might's attack when he got the Sludge Villain off me before he slapped me awake. I held on to his leg to try and stop him from leaving to ask him a question and ended up on a rooftop when he jumped. He said a quirkless person can't be a hero, to give up and find a more 'reasonable' dream."
"That lying, hypocritical, bastard!" his father roared in fury, "Nobody makes my boy cry! NOBODY! Inko, Izuku, one of my…coworkers, has a warp quirk. I'm sending him over to bring you both to me, we have a lot to talk about." Moments later, a swirling mass of black and purple smoke appeared in their living room that formed into a well dressed figured made of that same smoke.
"Greetings Midoriya clan, I am Kurogiri." The figure bowed politely, "I was instructed to bring you to my employer." Sharing a confused and nervous glance with his mom, Izuku held her hand as they walked through Kurogiri's warp gate and into a laboratory like room. In a chair attached to dozens of life support machines, most of his face clearly destroyed, Izuku saw his father, only recognizing him due to a birth mark on his neck.
"Dad!" Izuku rushed over, horrified at the horrible wounds inflicted on his father, "What happened?! Who did this?!" rarely felt rage started surfacing in Izuku at the second question, his mother sobbing as she held his father's hand.
"That's part of what we need to talk about Izu." His father's masked head turned towards him, "To start with, part of what you know of me is an alias. My birth name isn't Hiashi Midoriya, even if it is the name I've chosen to use. My true quirk isn't fire breathing either. In reality I have a quirk known as All for One, that allows me to take the quirks of others and use them for myself, one I've been using to take quirks for centuries."
"C-c-centuries?!" Izuku's mom gaped in shock, although having just been told by her husband that he was older than anyone else alive made that shock more than a little understandable.
"I was one of the first quirk users." His father nodded.
"But why lie?" Izuku demanded, his voice rough as he tried not to be angry at his father, wanting to know the reasons first.
"Because to the rest of Japan I was a super villain." His father's empty eye sockets seemed locked onto Izuku's emerald orbs "Japan was passing its first quirk use laws, forming the first 'heroes' and deciding people couldn't use their quirks ever unless they worked for the government at all times, were nothing more than attack dogs. So, I chose to rule the underground, the criminal element that would always exist. I intended to take over since those in charge clearly couldn't do anything right themselves. My biggest obstacle though has always been the betrayal of my own brother."
"I have an Uncle?" Izuku looked stunned, even as he listened to the story. He knew his father and could tell that he wasn't lying.
"Had. He was thought to be quirkless, so I gave him a quirk, one that allowed the ability to stockpile power. He didn't agree with my plan, believing the law was right and should be followed just because it was the law. Turns out he had a quirk after all, one that allowed quirks to be passed down. It merged with the power stockpiling ability into a new quirk. He attacked me, tried to stop me, and I was forced to fight back. He then gave his quirk with another to have them oppose me and fought me until he died as his power gave out. Each of his successors has come at me again and again and the most recent one, All Might, is the reason I have these injuries. All Might, the same man who was once a quirkless nobody that wanted to be a hero and was trained by the number one pro of his time." Hearing that, something snapped in Izuku. Knowing his idol, the man he'd poured his heart and soul out to, was just as quirkless as he was once and still told him to give up? It made him want to scream.
"So now my boy, I intend to let you prove him wrong." His father smiled as well as he could, given the damage suffered to his body, "I have a list of quirks I possess, as well as those I know of in the criminal underworld. Pick one and I'll give it to you. You can do whatever you want with it, even be a hero if you wish to." And Izuku could tell that his father meant it, the man who hated the hero system would let him follow his dream if he chose to go there. As he thought it over though, Izuku realized just how right his father was. Quirk laws were absurd, people could be sent to jail for using their quirk to defend their own lives in a villain attack! If someone used a simple telekinetic quirk to get a can from a high shelf at a store, they could technically get a fine if not jail time. And then there were places like his school that ignored those laws and allowed people like Bakugo to kick those with weaker quirks or no quirks around like soccer balls.
"Dad, I want to change this country." Izuku's tone was iron as he found his resolve. He wouldn't let people suffered like he had under the unjust laws that 'heroes' like All Might protected and enforced, a society where people like Bakugo were praised and seen as pillars of society for no more reason than they could destroy things.
"That's my boy." The pride in his father's voice made Izuku smile for the first time since All Might had left him on the room.
Groaning, All Might rubbed a hand down his face as he sat with Principal Nezu as well as Sir Nighteye in one of the UA offices, the three of them discussing potential candidates to receive One for All. "I'm telling you, Mirio is the perfect candidate." All Might's old sidekick insisted, "He is the ideal candidate to carry on your motto and take your place as the symbol of peace, always with a smile to uplift everyone!"
"I find there are also several good candidates amongst the incoming first year applicants." Nezu spoke up, "One Kirishima Eijiro for example has a hardening quirk that would let him better withstand the backlash of All for One and the young man is apparently already dedicated to trying to protect others and be a good role model. His idol he's based himself on is the Chivalrous Hero Red Riot according to his teachers."
"I-" All Might wasn't able to finish the sentence as the door to the office burst open to reveal Eraserhead.
"Aizawa, what on earth-" Nezu began only to be cut off by the underground hero quickly turning on the television and directing them to watch as a group of four appeared on screen. Instantly, All Might had to hold back a shocked rush of blood flying from his mouth as he recognized the quirkless kid from the Sludge Villain incident sitting at the center, staring seriously at the screen. Flanking him were a somewhat dumpy woman, a man whose body seemed to be made of smoke, and someone all Might recognized as a fully healed All for One.
"Greetings, people of Japan." The kid spoke formally and coldly, gloved hands steepled together as he looked at the camera, "I've come before you all today to…share a grievance I have with our dear country. It's quirk laws."
"What is this?" Nezu hissed to Aizawa, the principal recognizing the ancient villain as easily as All Might had.
"Nobody knows, but they've hacked broadcasts all over the country." Eraserhead grunted out as the child continued speaking.
"In our country, there are quirk usage laws to keep the common citizen from using their quirks, but why? I understand not allowing crimes to be done with them, but to arrest someone for using it casually? To punish a person if they use a quirk to defend their lives against a super villain attack?" standing up, the child walked around the table he was at as the camera turned with him to show an unconscious man that All Might recognized as a serial killer known as Muscular. "This man earlier today was attacking a populated area and killed several people who were unable to fight back for fear of being punished by the government for the crime of wanting to live. I intervened before he could kill the water hose duo and now he is here, to face my judgement."
"His judgement?" All Might frowned as the child took off his glove and approached the man, "He's quirkless, what could-" at a touch, the serial killer was blown apart before the pieces and blood swirled, reforming moments later back into the man who didn't look any different apart from waking up."
"All for One gave him a quirk." All Might grit out as Muscular fully came to, screaming in rage at the child and lunging for him, only for the man's quirk to not activate.
"I have taken you apart and put you back together without your quirk." The child glared coldly at the serial killer before knocking him out again, "Our country would let bastards like this live in prison to escape again and again while punishing civilians who's only crime is wanting to live and see their families. I have had enough of corrupt laws and false heroes defending those laws. If the law is unjust then it is our duty to see it changed, and I will do it myself if I have to, tearing down this broken society one brick at a time."
AN: So yeah, Izuku was given Overhaul in this and actually gets creative. We know it can be used to reshape the environment and people but it's never given time to really show what that can do besides making big monsters or blowing people up with the occasional instance of putting them back together. We see Overhaul use it on someone and actually cure every ailment in them meaning he can make them different when he reassembles them. In this, Izuku learned anatomy and biology to not only fix his father's injuries, but to also be able to disassemble people and reassemble them without the quirk gene. The thought of him doing that sounds terrifyingly badass to me.