Oh he definitely wants to go somewhere to get games. That’s been evident from what both he and Klopp has said. He agreed to stay last summer because Klopp asked him to.
He’s trained his entire life to play as a professional footballer. It’s not as simple as saying he should feel satisfied because he gets to train with world class players, earns good money and only has to sit on the bench. Money will no longer be a motivating factor for him anyway, as he’s already makes a great living and can also earn great money elsewhere. Playing games in front of 10s of thousands every week has to be something he strives for.
If I were him, playing as well as he did during a period when the club was suffering as it was, but to then see a younger player signed up to take his position, and then to have an always injured player also start ahead of him in games, I would want out. Especially after the club begged me to stay in the summer. Don’t blame him for pushing for a move.