The thing I'm wondering about is if Burnside really had the manpower to get clear to Raleigh NC. Checking the consolidated report for June 30 1862 the Confederate Dept. of North Carolina has 21K Aggregate Present and the Dept. of SC and Georgia has 30K Aggregate Present; IIRC a fair chunk of the Dept. of NC was concentrated around Richmond at that time, but if McClellan doesn't have the manpower to make a new offensive some of that can go back down to NC.Burnside and the Navy worked well together in the Sound in early '62 (Roanoke Island for example). This potent force in the side of the Confederacy was only broken by Lincoln sending Burnside to McClellan.
If the eastern railroad line had been broken, Richmond and Lee would have to rely on Virginia itself and the weak railroad line through Knoxville.
At the same time Burnside had 15.5K AP and Hunter had 21.5K AP, which means that both Confederate departments outnumber their Union counterparts; this isn't as much of a risk while operating near navigable water, but the Neuse river simply doesn't seem feasible as a power projection route up past Goldsboro (where it's about fifty yards across these days).
A Union capture of Raleigh would mean going another 45 straight-line miles inland from Goldsboro; I suspect at that point you're talking about needing serious reinforcement to Burnside's department, and it's back to the manpower argument again.
Had the Union continued recruitment through the spring of 1862 and had an ongoing manpower pipeline, on the other hand, all these constraints pretty much go away. Reinforce McClellan with another 20K new troops so he can keep most of Lee's force penned up in Richmond, give Burnside enough troops to muster 50K PFD and cut inland, and have Pope's Army of Virginia threatening along the line of the Rappahanock and rebuilding the Fredericksburg rail line; Lee at this point has too many cups and not enough water to fill them all.