Anon, let's totally forget the canon. Welcome to my land of creations, where I like to explore the possibilities within two characters interacting.
With that already clear, let me tell you that I don't understand why there's no more content about them.
They both are way too smart for their own good. They learn quickly from their mistakes, they adapt easily into new environments, they know how to create simple strategies and go with the flow. They are also the victims of a system that categorized them whenever they wanted and mostly by their looks.
Hawks comes from a criminal background (something about his family) and Shigaraki comes from a hero background (same case). Still, they turned Hawks into a hero and Shigaraki into a villain. They rescued Hawks be aus he was pretty and had an useful quirk, when Shigaraki was more spooky and his quirk was destructive.
They lost their innocence violently. They're trapped in the hands of bigger players and still they are so strong and smart and just— they would be not doubt a menace of a duo. They know too much. They are able of even more.
They'd fight and hate each other and betray each other. They would have this amazing sexual tension and stuff like "pretending to be a couple or protecting the other in my environment". Hawks would be like "I can make them arrest you right now" and Shigaraki would just smile. Yes, he would, if Hawks didn't need him. They both know it. Hawks wants to punch him, but somewhat he ends up smiling too because this little piece of trash. Shigaraki would just shield him from other villains and you'd have Dabi fuming over "your new pet" and the League just ships them. That's the most fun they ever had.
Enemies to lovers? With mixed backgrounds and symbolisms? The annoying possibility that maybe you care about him?
This is a 8/10, anon. The others are just blind. This is a hidden treasure waiting for a worthy writer to dare dig for it.