I don't think a failure to shoot is a significant problem. RVP drops deep and takes long shots, Nani when he plays loves to drift inside and shoot from the edge of the box, Young too when he's playing well, Cleverley likes his first time effort as it comes back from the box (even if he's not very good at it). Rooney should be in that list but he rarely shoots from range, and when he does he doesn't put his foot through it enough, looks for the pass and then the placed shot or even lob before just smashing it.
The issue with the midfield is fairly obvious and others have said it...unless the pairing is Carrick and Cleverley then it's pretty unreliable. A player in the mould of Dembele should be the biggest priority.