Let me turn the question around: when the Union armies in the west were clearly winning in 1862, capturing Nashville, Corinth and Memphis, why did Lincoln not order them to head east and reinforce the Army of the Potomac? It’s simple: the Confederate armies of the west remained a potent force and there were more objectives to accomplish (Vicksburg, Chattanooga, Atlanta, etc.).Surely though the reason IOTL that it was relatively rare to see troops from the Army of the Potomac sent west was because the Army of Northern Virginia was still intact and Richmond was still in Confederate hands until the very end of the war and thus those Union troops were still needed in the east. In a timeline where the Union takes Richmond and smashes the AoNV in 1862 though the situation is very different. There will be much less need for Union troops in the east in such a timeline, so why wouldn't Lincoln start sending troops from the Army of the Potomac to the western theater?
For a more modern analogy, should Eisenhower have redirected his troops in France to Italy after Falaise?
The only time I could ever see transfer of troops from the Army of the Potomac to the western Federal armies is if a crisis occurs. The Army of Northern Virginia might have been bloodied and Richmond has fallen, but the rebel army is still standing in North Carolina and has the capability to counterattack. Why not keep the Army of the Potomac fighting on to finish the job? It’s only when an army has accomplished its objectives that elements were redeployed.
As an example of an army that had chunks removed after victory, look at Grant’s Army of the Tennessee post-Vicksburg. With the Mississippi River cleared of Confederate outposts, Grant’s army did not have a mission any longer. Grant sent telegram after telegram to launch a Mobile Campaign, but instead the army assembled for the Siege of Vicksburg was almost disbanded:
- IX Corps returned to Kentucky for the Knoxville Campaign
- XIII Corps went to join the Army of the Gulf, which needed more manpower to defend its territory
- Elements of XVII Corps were detached for operations in Arkansas before being recalled for lifting the Siege of Chattanooga
Or in another case, the Army of the Gulf was practically disbanded after the failure of the Red River Campaign because their troops were better used elsewhere.