As we all know, more often than not, the Army of the Potomac decided to commit to a overland campaign in Virginia. This meant the Union Army would have to cross numerous rivers that went from west to east, which provided the Confederate Army a major defensive bonus. At the same time, the Union army would always send a small force to try and capture the strategically important Shenandoah Valley, however this would usually result in failure.
So here is my question. Would it not have been smarter to leave a semi-large (say 30 to 40 thousand men) to guard D.C. and everything North of the Potomac while sending the vast majority of the Army of the Potomac into the Shenandoah Valley? Doing so would not only provide ample protection for the Union capital, but it would also allow the Union army to use the bulk of its force to capture the strategically important valley, as well as bypass every single major river in Virginia and approach Richmond from the west instead of North (thus preventing the Confederate army from using its defensive terrain). Does this not make sense?