I think part of White not being used at WR has to do with White not wanting to be a WR.....yet.
Pre-draft, he was asked ad nauseum about whether he'd be ok playing WR. And he'd give the standard "whatever gets me on the field" followed by "but my heart is set on QB." The Dolphins notoriously told White they wanted him as a QB, even while they probably had other/additional plans for him. It's probably a bit too early for the staff to say to him "hey, ya know how we said you'd be a QB for us? Scratch that, you're now a WR."
I think they're giving him time to either shine or fail as a QB - he'll probably figure out for himself that he's undersized for a traditional QB role and much better suited to play a different and varied position, including some WR. But until he's ready to do that himself, he won't put the time in to succeeding at those skills. If he's going to be a WR, he needs to be practicing and studying running routes and working with the JUGS machine.
On a sidenote, I'm still not sure why we haven't seen him be the motion/jet guy on the Wildcat...he'd be running to the outside or throwing on the run...both play to his strengths, as opposed to trying to run up the gut and getting crushed by the pass rush. Maybe it's coming.