Lincoln, Gen. Winfield Scott retired, the general staff and field commanders were more interested in precluding CSA getting Lee simply because Lee was wanted by his bud Jefferson Davis and because Lee was from a prominent Southern family that drew from "Light Horse" Harry Lee in the war of Independence.
Everyone knew Lee from the Mexican American War where Lee the engineer cleared roads ahead of Scott's army and scouted to include some minor engagements. Grant the Infantry officer made his name in Mexico taking command of cavalry units whose colonels had been shot or lost in the melee. When Grant became general in chief for Lincoln he and his soulmate Sherman met on the road in Cincinnati, at the palacial Burnet House where Grant told Sherman to go get Gen. Johnston and Atlanta while Grant took out Lee.
Grant knew Lee and he knew Lee was average as an Army officer and that Lee was a humorless statue who when outflanked became enraged and attacked the center. True to form Lee at Gettysburg couldn't turn Meade's flanks and went to his default fault, the Union center on an open field at Cemetery Ridge where Gen. Hancock of New York commanded 8000 Infantry and a myriad of new cannon with new ordinance that obliterated three of Lee's divisions between breakfast and lunch. Lee galloped Traveler to Picket, "General, take your division and establish a defensive position on that ridge behind us." Pickett could only weep, "Sir, I have no division."
Lee's strong point to Lincoln and Gen. Scott was not as a some kind of gentleman or nobleman but as an aggressive commander who when the had the numbers prevailed consistently. With Lee commanding the Army of the Potomac it's likely the war would have ended much much sooner than it did with Lee in command of the Army of Northern Virginia and its inferior numbers, supplies, resources. Many agree Longstreet would have commanded had Lee gone with the Union which damps down enthusiasts of a quicker Union victory under Lee. Longstreet after all was right at Gettysburg on each decision where Lee was wrong and wrong grotesquely while overruling him. Pickett and Longstreet agreed at Gettysburg while Lee overruled 'em both on everything. Until Gettysburg Lee wanted a Caesar style battle of annihilation against the Army of the Potomac which he could never achieve while losing troops he could never replace in adequate numbers until the numbers ran out. Lee's casualty rate for the war exceeded Grant's.
The final note here and as it was without Lee, Lincoln and Gen. Scott knew they were looking at lesser commanders starting with MacClellan. MacClellan and Halleck translated Napoleon for West Point and Union war officers but in the field neither of 'em could hold Napoleon's horse. Lee and Grant relied heavily on Napoleon's flanking maneuver but of the two only Grant could execute it properly and successfully. Lincoln made Halleck "The Brain" Quartermaster which kept the supplies flowing until game over.