There's a great piece in Civil War Times by historian Michael Fellman arguing that Sherman's flagrant bigotry amounted to insubordination. Here's Sherman offering his views on colored soldiers:
I like niggers well enough as niggers, but when fools and idiots try and make niggers better than ourselves, I have an opinion.
You can about guess as to the nature of that opinion. In Ken Burns Civil War doc, historian Barbara Fields argues against giving people an "of their time" pass, because often there are people in that time who are on the right side of history. You can't at once credit the enlightened, without calling out those who stumbled around in the darkness--especially those who did so willfully.
I think that really holds true of Sherman. To an extent Sherman was of his time, as there was no shortage of racism in the Union Army. But by the time, Lincoln is pushing Sherman to recruit black soldiers, many of Sherman's peers seem to have been radicalized by their contact with actual slaves and actual slave-holders. Apathetic attitudes toward slavery functioned a lot better, when one's contact with plantations was minimal and planters weren't sending whole villages off to die for the right of property in man.
Grant starts off as a slave-holder--though one who manumits--then later argues for blacks in the Army, still later fights the Klan, and then almost literally on his death-bed dismisses slavery as one of the worst causes any nation ever fought for. ("By arming the Negro we have added a powerful ally. They will make good soldiers.) George Thomas (ironically a sometime-foe of Grant) begins as the scion of planter family in Virginia, who had to hide out as Nat Turner rampaged across the countryside. "Old Pap," though a Virginian, fought for the Union during the war. Here is Thomas after the Battle of Nashville, as depicted in Christopher Enolf's biography:
Thomas had supported the recruitment of African American troops, but thought they lacked the courage and discipline to make first-rate soldiers. At Nashville his lack of white troops forced him to use African American troops in battle...
On December 17 Thomas rode over the battlefield with James Steedman and Thomas Morgan, reviewing events of the previous two days. He had not been to this portion of the battlefield on December 15 and 16 and had not seen the African American soldiers in action. What he saw on December 17 affected him strongly.The bodies of black Union soldiers lay in heaps along the battlefield, mingled with the bodies of white soldiers who had also participated in the assault. Many of the corpses lay in piles before the Confederate works...Turning to his staff officers [Thomas] said, "Gentlemen, the question is settled; Negroes will fight."
Fun fact#1: The "Negroes" were only there because Sherman left them behind. When Sherman invaded Georgia, he took what he deemed to be his best soldiers, all of whom were white.
Fun Fact#2: I first encountered Sherman as kid reading up on the origins of the 40 acres and a mule myth. The myth is based in Field Order Number 15, which was supposed to give land to blacks for resettlement in Georgia, South Carolina, and Florida. The author of that order was William Tecumseh Sherman.