I think it's more of a matter of adapting to a new league more so than a specific club. He excelled with you guys in a league that isn't exactly lightweight or pedestrian, but this is a league that is an incredibly physically demanding and high intensity league. He has no problems adapting to the intensity aspect of it - he did after all play in a very youthful and exuberant Dortmund team - and he has one of the best stamina levels within the squad. I honestly can't individually criticize him too much though, he needs to be more imposing with the ball and not shy away from responsibility, but he's playing in a team that doesn't allow him to flourish.
The Premier League is embracing more technically-gifted and slight players and the general tactical and technical standard of the league is improving every year, so his physique shouldn't be an excuse however, but I believe United should be adapting to technical players like Kagawa and Mata and not vice versa. It's a waste of their talent if we continually persist with out-dated wingplay that doesn't allow creative players to express themselves and doesn't allow them to combine in central areas, such is our static movement and rigid system.
He could really be a big hit here under the right tactical guidance and management. Which poses another question of our mis-management of him. Like you've said, we haven't allowed him to flourish in a set role when he hits form. I would preferably like him to play in offensive midfield, but it's unfortunately understandable that that would be difficult when we have Rooney as a supporting striker and a world-class striker in van Persie.
When it comes to his performance for us, it's more to do with the system than the fixed position however. He's flourished on the left for Japan, so he's complete adept at playing there given the correct style. I would like to see him start regularly now with more emphasis on creating in the final third and him playing behind RvP or Rooney and Mata on the right, as he's more adaptable to play wide.
For the sake of his career, he probably should go though. I'm sure he would be a big hit at another club.