Your "Christmas present" figure is the best description of Sherman's contribution. Certainly the defeat of Hood's army outside Atlanta and Hood's decision to burn military stores rather than allow them to fall into Federal hands helped in that it made the "peace platform" of McClellan and the Democrats seem a more dubious proposition, in that it helped to create the impression that the South was on the ropes. Any intelligent voter, though, should have known the South was on the ropes after the defeat at Gettysburg and the capture of Vicksburg the following day in July, 1863.
Sherman's "march to the sea" campaign, his boast that he would "make Georgia howl" and his vindictive destruction in South Carolina occurred after the election. In fact, from a purely military point of view, Sherman abdicated his responsibilities when he decided to march on Savannah rather than to pursue Hood and the Army of Tennessee. As matters stood in November, 1864, Meade's army, directly confronted the Army of Northern Virginia in its lines at Petersburg and Richmond, and Sherman was opposed to the Army of Tennessee south of Atlanta. Their respective tasks were to destroy the armies each of them faced. Sherman abdicated that responsibility with his relatively, militarily useless march to Savannah and his revenge march through South Carolina. The Army of Tennessee escaped to fight another day, and George Thomas in Nashville was left to face Hood, who rather quickly responded to the freedom Sherman's departure accorded him to attempt to "re-invade" Tennessee and acquire new supplies and new recruits. In the Trans-Mississippi, A. J. Smith was trying to run down the remnants of Sterling Price's Missouri State Guard, and the shattered fragments of Texas, Arkansas and Louisiana units which had attached themselves to Price. Price's "threat" was so minimal that when a genuine threat arose as Hood crossed the Tennessee River, Smith was sent to reinforce Thomas, whose Army of the Cumberland had been stripped of its best troops and nearly all of its transport animals by Sherman prior to his march into Georgia. Schofield held up Hood at Franklin, and Thomas assembled the forces necessary to defend Nashville. After stupid, useless frontal assaults on a nearly impregnable position at Franklin, Hood slowly gathered his bleeding, battered army on the heights south of the city. He had rather deftly forced the retreat of Federal forces on the Tennessee River by simply masking the position, and then crossing farther up river. Possessed by some inexplicable idiocy, he threw his troops headlong at Schofield's position at Franklin on the Harpeth River, rather than executing the move as he had on the Tennessee, which would have forced Schofield's retreat, and perhaps even have left Thomas unable to defend Nashville.
But Thomas was just about the best general officer in the Federal army in that war, and he had assembled every man, horse and mule he could strip out of Tennessee and Kentucky, and reinforced by Schofield (who had retreated in good order) and A. J. Smith, he finally had an even larger army than Hood. He held off his attack until he could assemble sufficient horses and mules, even using his precious draft animals, because he fully intended to destroy Hood's army. A typical mid-South ice storm forced him to hold off his attack for three days, and Grant had actually drafted a telegram relieving him of command, but which the senior telegrapher at the War Department had quietly refused to send. When the weather cleared, Thomas attacked, and his infantry reserves, mounted on the horses and mules he had scraped together, pursued the remnants of Hood's army until they were scattered and no effective formations remained. Hood's outright casualties were about 6000, or about 20%--but the battle started on December 15, and Thomas' forces pursued Hood until the rump of the Army of Tennessee recrossed Tennessee River on Christmas day. The destruction might have been more complete and the casualties higher had N. B. Forrest not fought desperate rear-guard actions to delay the pursuit.
Hood resigned in January, 1865, and was never given another command. Grant (who didn't like him) continued to ignore the contribution of the Virginian, George Thomas, who had accomplished what Sherman should have done and failed to do. The remnants of the Army of Tennessee, of which at most 10,000 survived the Nashville campaign, filtered over the mountains to join Joe Johnston's army facing Sherman in South Carolina. This fragment of the Army of Tennessee was the core around which Johnston built up his army to confront Sherman, but it was scattered forever in the aftermath of the battle of Bentonville, North Carolina in March, 1865.
Sherman's march to the sea and the path of destruction he cut through the Carolinas had great dramatic appeal, but the real yoeman's work was done the the little armies in Tennessee under Thomas' command--the Army of the Cumberland, the Army of the Ohio and Smith's XIV Corps--while Sherman played for the headlines.
FM will want to read up on Andrew Jackson Smith, who defeated Sterling Price, the boogie man of the Trans-Mississippi and
who defeated N. B. Forrest at Tupelo (Forrest being the Great Satan of the western theater). Smith was born in Bucks County, Pennsylvania.
I doubt that anyone will ever be able to convince me that Sherman served any real useful purpose--and certainly not crucial purpose--after he marched away from Atlanta.