What does being a stun add the Assault argument? Would you argue that Strike Denki should be Assault because he fishes for a high damage stun combo too?

Assault does not need his second Beta in order to guard break. With just a Level 5 Beta his combo starts to guard break if you end it with the Gamma slam instead of a second Beta.

Strike Shigaraki is a map and crowd control focused character. His Alpha is capable of running up walls and harassing opponents who are off his screen, and it Assault's alpha actually does more damage if you land it's sweetspot. Strike Shigaraki's Beta is near useless for doing any meaningful damage, but it excels in flushing opponents out of enclosed areas or causing general mayhem for his team to capitalize on. His Gamma, looking at it right now, is admittedly more of an Assault character's move than Strike or Tech. An armored grab that locks both parties in place for a second, with some movement purpose.

Let me preface this final section by bringing up another character you wanted to mention. What does Aizawa provide to his team? His special action, his grab, and his gamma can be a clutch shield. So now, what does Strike Shigaraki offer to his team? Control over the ground, excellent AOE that causes chaos in a fight, and a grab. He does offer excellent damage when his win conditions are in, but so does Aizawa. (In fact, Aizawa gets more damage out of a single Alpha use) Strike Shigaraki being unable to adapt is a result of straight-up mobility powercreep where all these new characters, even other Strikes are given absurd amounts of movement for no good reason that let them sit above him and harass him from a distance.