During October 1862, McClellan was told to give his plan for an offensive, and his initial plan (as told to Halleck) was to invade the Shenandoah, going after Lee's army directly. His specific objective was Winchester, and he planned to do this because he would be able to either force a battle with Lee's army or force Lee out of the northern part of the Shenandoah - and either outcome was of benefit. If he forced Lee out of the lower Shenandoah, it was no longer the case that Lee could invade the North the moment McClellan moved away from the Potomac.
This plan was neither approved or rejected, but during this period Halleck and Lincoln both tried to persuade McClellan to opt for a different plan - a plan to move south on the east side of the Blue Ridge mountains.
Lincoln's reasoning was that doing it this way would let McClellan move to Richmond before Lee could get back there, and thus take Richmond without any need for a fight.
Halleck's reasoning was almost directly opposite, arguing that doing this would help protect Washington from a Confederate army on the Rappahannock. (Note that this army did not actually exist.)
I am not actually sure why neither man simply ordered McClellan to take the eastern route. They had the authority to do so, but instead the only order actually given to McClellan was to submit his plans for approval.
In any case, the deadlock was broken in late October when McClellan opted to follow the "eastern" plan - and, unlike his original offensive plan, this got approval within the day. At this point McClellan began bridging the Potomac; on his move south he took pains to operate his corps so that he had a significant force advantage (in reality) at any point where a battle might break out. For example, AP Hill's strong six-brigade division was on the western side of Snickers Gap, and McClellan moved to take the gap on 2nd November with both 2nd and 5th Corps available if a fight developed; meanwhile DH Hill's strong division was south of there, at Upperville (on the eastern side of the Blue Ridge) and 1st and 9th Corps were deployed so as to be available if DH Hill moved towards the Union army in concert with AP Hill.
On the following day, 5th Corps held Snickers Gap to block AP Hill, while 2nd and 9th Corps concentrated against DH Hill, driving him back over the Blue Ridge without need for a fight, and on the 4th of November 2nd Corps occupied Ashby's Gap while 9th and 1st Corps moved south.
On the 5th, 9th Corps took Manassas Gap, while 1st Corps siezed Warrenton.
Consequently there are "jobs" for at least four corps during the advance - three corps holding a gap each, and a fourth to take Warrenton. This defines the force which has to be over the river to make the historical advance possible while blocking Confederate interference
It's possible that those four corps could have been moved so as to be over the river more quickly, with better time and motion. I don't argue that McClellan's movements were optimal. But of those corps, one (5th) was covering part of the upper Potomac against the possibility that Lee was going to invade the Union again, and this corps began moving to cross the river starting on the 29th (it reached the crossing place on the 30th, spent some time resupplying to cover the supply deficiencies of the previous month, then was marching south from the 1st of November onwards having crossed the river.)
That same bridge was in use for 2nd Corps before 5th Corps crossed, and they crossed effectively consecutively. What this means is that it looks like to get those four corps over the river and moving south you need to speed up all the operations at Harpers Ferry (meaning 2nd Corps has to cross earlier to clear the bridge earlier), and you also need 5th Corps to be able to move away from the Potomac sooner.
Historically what freed 5th Corps up was confirmation that Longstreet - who had been at Martinsburg for the previous month, within a day's march of the Shepherdstown fords and not much further from Williamsport - was moving south. This confirmation came late on the 28th, after Longstreet broke camp the same day.
Part of the reason why I've outlined this is to make clear what at least some of the constraints on McClellan's movement south were. In the event, he could use 5th Corps (and then 6th) because Lee was abandoning the possibility of threatening a renewed invasion of Maryland, but this change of posture by Lee happened only once McClellan had already begun moving.
Absent this, for his manoeuvres McClellan would have had to rely on having only three corps available in Loudoun Valley, plus a reinforcement column out of Washington promised to him, and after events of March, May, June and July we can perhaps see why McClellan would be a little leery of believing in such promises - but he was on the move anyway while he couldn't necessarily rely on 5th or 6th Corps.
I sort of wonder if McClellan's intent was to provoke that reaction by Lee, so he could have the troops he needed to do the tasks he was going to need them for, but that's impossible to prove one way or another.
But another part of the reason is to outline why Lincoln's suggestion - that McClellan could march all the way to Richmond before any substantial part of Lee's command could block him - doesn't really work. The time taken to cross two corps over a river, nose-to-tail, is about three days simply because of the length of the various columns of wagons (without any allowance for resupply etc.) and Lee started Longstreet moving at a point when only one of McClellan's corps was even south of the river. At that point, over half of Lee's army (the whole of Jackson's corps, plus Walker's division that would be part of Longstreet's corps) was already closer to Culpeper than any part of McClellan's army, and they'd been in those positions pretty much since mid-October. The other half of Lee's army (Longstreet) was about 1.5 marches (15 miles) further from Culpeper than McClellan's most southerly divisions on the morning of the 28th - and on the morning of the 28th, all but 9th Corps were still north of the Potomac.
What this means is that there is no component of McClellan's army which can get to Culpeper, from a movement starting on the morning of the 28th, than most or all of Lee's army.
So at the time McClellan launched on his move south, he was anticipating the possibility of having to fight at least half of Lee's army, if not all of it - either in the Loudoun Valley or at Culpeper in a blocking position. In the event, Lee made a mistake, which is that he moved Longstreet to Culpeper (and got there first by a considerable margin, because he didn't have to manoeuvre, wait for trailing corps or advance to contact and could just route-march) but left Jackson in the Valley. This would have been a good move had McClellan left his flank open to raid his wagon columns as he moved south, but McClellan effectively covered his flank by blocking the mountain gaps and so Jackson had nothing to raid.
The true magnitude of Lee's error doesn't really manifest until about the 6th or 7th of November, though, because this is the point at which McClellan's army is now supplied (out of Washington) and within striking distance of Culpeper. It is now impossible for Jackson to raid McClellan's supply lines - there's the Bull Run mountains and a whole separate corps in the way, even if he could march forth from the Blue Ridge in that direction - and the only safe means by which Jackson can join with Longstreet is through Swift Run Gap, which is what Lee is telling him urgently to do.
Jackson isn't moving, though, whether because of personal preference or massive supply problems that would ultimately take him weeks to resolve (since, either way, he didn't move for another couple of weeks).
This means that Lee's movements - and the way in which McClellan has operated his own corps, so as to avoid giving Lee an opening to exploit rather than getting himself strung out trying to win a foot-race to Culpeper - have created the situation where Jackson is unable to support Longstreet, and Longstreet is unable to fight McClellan's army by himself. Lee showed a marked reluctance to try, in fact, preparing to retire to Gordonsville.
This is not a good situation for the Confederates to be in, because:
- McClellan was moving south, and was willing (we know this) to attack most of the Confederate army at Antietam with at least three quarters of his own. Here he is facing half the Confederate army and his own army is much stronger than at Antietam, while he knows he inflicted heavy casualties at Antietam.
- Lee was going to get out of the way rather than fighting McClellan at those odds.
- With the Rapidan effectively undefended, McClellan can either swipe at Gordonsville or move to Fredericksburg. Going for Gordonsville gives Lee a choice between separating Jackson even more thoroughly from Longstreet (by moving S/E) or moving west to reunite with Jackson through Swift Run Gap - and the latter means that there's literally nothing between McClellan and Richmond but distance.
- By moving to Fredericksburg, McClellan can get himself in supply south of the Rapidan, and from there outflank (without needing to fight) any Confederate defensive position short of Richmond.
- Lee simply has no good solution to this. He needs Jackson to unite with him to be able to fight a pitched battle with McClellan (i.e. one where the majority of both sides are actually involved in the fighting), he has no means to effectively generate a manoeuvre victory (he can't knock out McClellan's supply lines) and there's nowhere he can force McClellan to attack over a strong geographic defensive position.