The fact is, United's system works in England very well because of the way they defend and counter attack. United challenge anyone and everyone to beat them with technique. Since most teams in England are more focused on running fast and tackling hard, the players forget or never learn how to play a highly technical game. As a result, United win a lot of defensive battles, leave their wingers and strikers high up the pitch, and counter attack you to death.
If they are forced into playing a game when they are stuck work with 10 men behind the ball, they push everyone forward as much as they can and they attack you with their wingers and strikers. They want you to earn their respect. They test the defenders all game long. Not only do they test them and attack them, but they attack them in different ways. If you compare Downing to Valencia you'll see Downing will run straight down the wing and then cross it in. Valencia runs AT the defender, beats them, and crosses it in. The difference is night and day on how it creates defensive movement. The wingers are such a key part of United's success in the league it's ridiculous. Why this works not as well in Europe is very simple. Most of these players are used to technical players. It doesn't bother them one bit that someone is trying to do a few tricks. That's what they see every week. Even against the worst sides they are trying to beat them with technique. So when Nani tries to do a few tricks, the defenders simply don't care.
The midfield play a very simple role in this system. Support the attack. Support the defense. Don't make big mistakes. If it's time to defend, the central midfield better defend. If it's time to attack, they better run as hard as they can and support the attack. Their level of involvement in picking the key pass or stretching the defense is limited. This isn't to say they don't have to be quality passers, but it is to say they have very little responsibility in scoring goals, creating goals, or really doing a whole lot of tackling. Carrick is their defensive midfielder and he can't tackle. Period.
You can't be horrible to play in United's midfield, but you really don't have to be all that special. You just have to be a hard working, well rounded player, who can pass the ball pretty well. Central defenders, strikers, and wingers are the most important players on the pitch for United.