Izuku's quirk is Take. It's an emitter quirk that lets him take away other quirks. And that's it. He can't use them. He can't give them back or to other people. The only reason he knows he takes is because the doctor found a second quirk factor after it's manifestation, and it matches the missing quirk of the thug that attacked them. He can only take take take! Its why his father abandoned him and his mother. Saying "I don't want to live with two failures." as he leaves for good. It's why Kacchan hates him now, its why his teachers hate him. But he doesn't want to take. He wants to give too. That's his only thoughts, as he asks his idol that question apon that rooftop.
"Can I be a Hero, even if my quirk only takes?" His idol was conflicted after hearing about his quirk and what it does, but he eventually says Yes. He says he can be a Hero [After all, All Might realizes that Take is basically a glorified Erasure Quirk, just one with a permanent effect] And his life only goes up from there. After saving Kacch-Bakugo from the Sludge Villain, and after those 10 months of hell, when he finally gets One for All, suddenly he has a way to give. Although it only comes to him after he's shattered three bones and caved in the face of a giant robot, where the (apparently) haunted quirk tells him that One for All combined with Take in a strange way not unlike how One for All itself came to be, fusing together perfectly into one singular quirk.
He can now give the quirks he's taken by willing a specific quirk to pass on, and having them digest his DNA. Although he still cannot use them.
That is until Blackwhip comes out, and the sixth informs him of how One for All absorbed the essence of any quirk he's taken, and replicated it within its core, waiting there for The First to eventually give to him [Only once they find out Izuku's reaction to it. They don't want another All For One, especially since he has One For All. They can't see his memories and the only reason they felt comfortable with him being the 9th was because he couldn't use any quirk he takes, although after thier time with him he seems golden-hearted. The First knows that power corrupts like no one else however, after all that was what made All For One himself] Eventually, he feels great. He feels like his quirk isn't a curse anymore, that he does not take without reason anymore.
Or, the story of how a rather unfortunate combination of Attraction of Small Objects and All For One combine to make a quirk that doubles down on taking and nothing else, albeit at a range. The story of how All For One mistakingly creates the second part of his downfall, and of how a new, better One for All was created due to his mistakes yet again. The story of Take For All, the quirk that takes, but also gives, for the benefit of all.

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MHA Story And Quirk Ideas
FantasyStory/Plot and Quirk Ideas I have no motivation to make, but think sound really cool! The entries with Quirk : [Name Here] means this entry is about a quirk idea! The entries with Story: [Name Here] means it's a story idea! If they're combined, it m...