It's almost like bullies are universally relatable no matter what age you are since they are literally present at every stage of life. Whether it's a bigger kid in elementary school or a power mad manager at your job, everyone has interacted with a bully at one point of their life.

You're also slightly mischaracterizing Sadeas as just a bully. That's incredibly reductive. Is he petty? Yes. Is he vindictive? Sure. Is he cruel? Yup. Is he a bully? No, he is Alethi. He is doing what he can to protect the Kingdom in the typical Alethi Highprince way. Politics, maneuvering, and betrayal.

He views Dalinar as weak for following the codes and dislikes that Elkohar is seemingly following suit. The only way to fix that is to remove Dalinar. Hence, his focus on discrediting him to remove him from power.