Izuku was born without a quirk. That didn't change the fact that she still told anyone who would listen that she wanted to be a hero when she grew up. As she got older though, and watched what heros went through and felt the jabs of her classmates, Izuku decided that she didn't want to be a hero anymore. She also decided that just because she didn't have a quirk, didn't mean that she couldn't stand up for herself. Or others.
