My Hero Academia's Tomura Shigaraki has been a thorn in the heroes' sides since the beginning of the series. However, before he became such a notorious villain, Shigaraki actually wanted to be a hero as a kid. He admired All Might and other pro heroes. As a child, Shigaraki always dreamed about becoming a hero and saving others, but his father beat that aspiration out of him because of what happened to Nana Shimura, his grandmother.
Shigaraki once aspired to be a hero, but it appears that his Decay quirk predestined him to become a villain. There is no way to escape Shigaraki's Decay unscathed, unlike other My Hero Academia quirks that could be interpreted as villainous, like Hitoshi Shinso's Brainwashing. It guarantees the instant death of its victims, especially now that Shigaraki is more powerful. In the world of heroes, such a destructive quirk would be useless.
Many MHA Quirks are Dangerous, but Shigaraki's is Particularly Dangerous
It's doubtful that Shigaraki would have gained fame even if the tragic incident in which he accidentally killed his entire family had never happened. Without a doubt, his father would not have approved of his enrollment in U.A. High or any other hero academy. He would still be a young boy wandering the streets if All For One hadn't taken him in, drawing a lot of strange looks from adults and professional heroes who had more important things to do than assist a lost child. No matter which direction the story went, Shigaraki was clearly going to turn out to be a villain.
The idea of dangerous quirks isn't anything new in My Hero Academia. Some quirks are hazardous to their users, like Dabi's Blueflame or Deku's One For All overwhelming his body with such a massive amount of power. Other quirks are safety hazards for everyone else, like Shinso's Brainwashing, which traps its targets with a response or Thirteen's Black Hole. However, Shigaraki's Decay is in a league of its own. There's no way that it would ever be useful for heroic causes. It can kill with a single touch, and once Shigaraki touches a single object, it may have a domino effect and destroy everything behind it, killing everyone in its path.
Uraraka's Zero Gravity is useful on rescue missions because she can lift debris and rubble out of the way to make sure no innocent civilians are trapped beneath. Theoretically, Shigaraki's could be useful for similar reasons. The quirk could destroy a door to rescue hostages or break down particularly large pieces of rubble to rescue civilians. However, Decay is so destructive that one object simply isn't enough for it. The quirk craves destruction, and it was bubbling under the surface for years before it manifested in Shigaraki, hence his painful skin condition. Decay destroys one object, then it wipes out everything else around and behind it.
Since Shigaraki's attempt to seize power during the Paranormal Liberation War, decay has become even more deadly. Now it is capable of destroying vast swaths of land and tearing through cities as if they were made of sand. Decomposition has a rapacious propensity that constantly yearns for more destruction. The quirk also changed Shigaraki's psyche, making him equally obstinate in his pursuit of the greatest destruction. Even in everyday life, such an oddity would not be accepted in a heroic society.
Shigaraki Wouldn't Fit in the Heroic World
Shigaraki's father would have prevented him from enrolling in any hero academy or pursuing a career as a hero even if his quirk hadn't materialized without causing such a massively destructive event. He would be forced into the standard job market as a result. But in a society where looks are everything, Shigaraki probably wouldn't fit in because of the negative stigma attached to his skin condition. Once more, this would force Shigaraki to the streets, where he would probably ruminate on how society had rejected him for reasons beyond his control and end up with the bad guys.
If All For One never got his hands on Shigaraki, it's still likely that he would have become a villain. It's also possible that Shigaraki could have died on the streets as a stranded child with no food, shelter or anyone to look out for him. Shigaraki was basically discarded by society for things out of his control. Abused by his father and intentionally neglected by society because of his appearance, it's understandable why Shigaraki felt rejected by society and was eager to follow All For One's lead. It's clear that Shigaraki respected All For One as a mentor and even potential father figure. This makes All For One's betrayal of Shigaraki and the League's trust even more heartbreaking since he never cared about the vulnerable, scared kid he took in and molded into a supervillain for his own gain.
Even if Shigaraki had a chance of becoming a pro hero, he'd likely never make it into any notable position on the Hero Billboard Chart. Society is currently struggling to accept Endeavor because of his volatile past and connection to one of the most deadly villains associated with Shigaraki. If Shigaraki entered the hero market, he'd never earn anybody's trust or respect. In that case, he may have become a vigilante like Stain or even defected from hero society entirely like Lady Nagant.
Either way, Shigaraki wouldn't have a place in hero society since his very existence would challenge everybody to observe their prejudices and realize that it's easier to avoid someone like him than provide him the necessary resources to live like an average person. This does pose the question of whether Shigaraki would have turned into a villain if even one hero offered a hand out to help him, although it's unclear whether Shigaraki would have accepted it. Would he have welcomed any offer to get him off the streets, or would he have turned a helping hand to dust, perceiving it as a cheap display of pity? No matter what happened to Shigaraki, it seems like he was doomed to a life of villainy. Whether or not Shigaraki can be redeemed after regaining control of his body is currently unclear. However, even if Shigaraki atones for his actions, he still wouldn't have any place in hero society.