When these kids with "villain" quirks needed help/ went through these traumatic experiences, the Heroes let it slide whereas the League helped them and gave them a family.

1. Tomura Shigaraki

We all know the leader of the L.O.V, the crusty Shigaraki. He comes in as a perfect example of one of these kids. His quirk Decay causes anything he touches with all five fingers to disintegrate. Due to his quirk, he ended up accidentally killing his entire family and sending him into a state of panic. In the hero society, those with "villainous" quirks were made to be outcasts. One For All came and gave Shigaraki a place in society. Shigaraki accepted, not caring about his status.

2. Toga Himiko

Another perfect example of this is Toga. Toga's quirk, Transform can only be activated by drinking someone's blood. Due to this, she grew an interest in blood and organs. She was told from a young age that she was a freak and a monster. Because her quirk involved ingesting blood other than her own she became an outcast, hated by even her family. she was found by Shigaraki and given a home and a family that understood her.

3. Dabi/Touya Todoroki

Touya or better known as Dabi (following the theory that he was sold to the government: https://www.youtube.com/watch?v=bG8zm38IjOQ ) was sold by his father Endeavor to the government camp that Hawks was sent to. Later having escaped, feeling betrayed by his family he broke down in the wood, body covered in scars and burns due to his quirk not being compatible. He was found by the League and taken care of, cleaning him up and managing the best they could with his skin. Dabi joined the effort to kill the heroes with the promise of family and love.

Now those are just to name a few, but what the heroes don't see is the L.O.V are a family, not by blood, true but a family at heart. With the best intentions, the "Villians" love and take care of each other, which is more than the heroes ever did for them. Most of them wanted to be Heroes but due to society's cruel view, they were neglected, becoming what they are.

I hope you enjoy my little rant!