That's not what Adolin himself thinks of situation. I wanted to bring the quote, but someone already did it. Adolin doesn't want to be a King, because he does not want to make his life difficult, to take responsibilities and sacrifice his "free time".
Dalinar is not good at politics. He could find someone to do it instead of carrying all responsibilities alone, but he decided to be better, do beyond his comfort zone and learn something new, something even unnatural to him. And he did learn well. When Dalinar is asked why he is doing this he answered simply "because I need to". That's his duties. Even if he doesn't want/like it, which he clearly stated. This is responsibility.
Adolin never goes beyond his comfort zone and never even try. That's why he's never had any responsibilities in his entire life. Another good example is in the first book when he insulted Dalinar and shouted at him in from of his men, unintentionally forced Dalinar to think about abdication, then realized what exactly did he do and what does it mean for him. Once he realizes it means he's the one to become a Highprince and hold responsibilities (and no free time for wine, girls and magazines) he immediately begs Dalinar to change his mind. That's Adolin's "responsibility".