If you read Grant's memoirs, you'll see almost no mention of the battle, other than that there was one. He disposes of the whole affair in a couple of sentences, and then begins telling of the moves made by Hancock (i believe) which would force Lee out of the position. The initial assault saw 7000 Federal troops shot down in under an hour, and thereafter, the troops refused to advance to the attack again. These men were certainly not cowards, but they knew it was suicide. Many soldiers understand the principle of the "forlorn hope,' in which a certain number of people will be lost in order for the rest to be in a position to exploit the situation. Nothing like that was going to happen at Cold Harbor.
In Freeman's R. E. Lee
, he recounts the testimony of a Virginia Justice of the Peace, into whose yard Grant rode with his staff. Being a gentleman, and schooled in urbanity, he offered them some buttermilk, the only cold drink he had available. Grant ignored him, but according to the old gentleman, then took out his watch and said: "If i don't hear the Old Fox's guns in fifteen minutes, i've got him." According to that witness, ten minutes later, they could hear a heavy cannonade in the direction of Old Cold Harbor. The "Old Fox" had done it to him again.
It helps to understand the entire campaign to understand Grant's actions. It also helps to understand Grant. Grant was no kind of tactical commander. He certainly was no coward, as his record shows, but both he and his mind tended to wander in the long, often boring and confusing course of a battle. At Palo Alto, he just wandered away from his regiment (he was an infantry Lieutenant then). At the assault on the San Cosme gate at Mexico City, where the fight was hot and the necessity obvious, he lead some men to manhandle a gun up a church tower, from which he began firing on the Mexican defenders, and then rounded up all the men he could find and joined the assault. If he could see the need for immediate action, he acted. Otherwise, he seems to have lost interest in protracted disputes. At Belmont in 1861, after the Federal troops had overrun the Confederate camp, he wandered off on horseback, and when the Confederates counterattacked, he was off by himself, and three southern riders very nearly captured him--he rode out of the woods an up the ramp of the Navy transport still on the river bank, just barely eluding his pursuers.
In 1864, using his new authority, born of his prestige and the obvious fact that he was the man Lincoln was looking for, someone who "understood the numbers" (i.e., that the South could not survive a war of attrition), he assembled a massive army in excess of 100,000 men. He stripped the defenses of Washington, and called in garrison troops from all over the east coast to replace them. Then he came down on the Army of Northern Virginia like a ton of bricks. He expected them to fold, but that was a serious underestimate of the quality of the army he was facing.
On May 5, he sent his army south in two wings, the western wing marching south southeast on the Brock Road. Lee's army was necessarily dispersed, as he couldn't know where Grant would attempt to cross the river, but it was old, familiar ground, and he soon had two corps marching to intercept Grant's army. Over the course of three days, every time the Federal troops attempted to turn the Confederate flank, Lee would throw a corps of infantry at them, and they "danced" south on the Brock Road. The Federal troops would march south, and Lee would have just brought up more troops to confront their next flanking attempt. At one point, Longstreet was wounded, and Lee seemed to become a little unhinged (he has lost Jackson fighting over the same ground the year before). Anderson, now commanding Longstreet's corps, sent in the Texas Brigade, the only reserve he had left at the time. Lee took off his hat, and began waving it and shouting "Hurrah for Texas," and "I will lead you, boys." The troops began to panic, and began shouting "Lee to the rear, Lee to the rear!" NCOs grabbed his reins and lead his horse to the rear--i truly think he was attempting suicide by combat. The Texans then attacked with such ferocity that the Federal commanders thought they were being assaulted by an entire division, or even a corps, and the fighting ended for the day (May 6, 1864) . (Lee to the rear, at Encyclopedia Virginia.
Stuart had been on Lee's right, to guard against Federal cavalry getting into the rear of the army. That is exactly what the Federal cavalry attempted to do, and a nasty, bloody fight ensued on May 7 at Todd's Tavern. The Federal cavalry drove in Stuart's troopers (chiefly Wade Hampton's division), but they failed to cut the road to Spotsylvania Court House, which was where Lee was marching his army, to interpose between Grant and Richmond. Although a tactical victory for the Federal cavalry, with heavy casualties on both sides, Lee had managed to elude Grant once more, and had accomplished his purpose of interposing his army between Grant and Richmond.
But the nasty fight at Todd's Tavern had delayed Lee's arrival at his new line, and the army filed on to the high ground and began digging in, leaving a huge salient they dubbed "the Mule Shoe." Grant had with him a brigade commander, Emory Upton, who would have a profound effect on United States military policy many years later. He advocated attacking an entrenched position with columns of infantry who would not stand and trade volleys with the enemy, but who would rush on the enemy position regardless of the cost. On May 12, they came howling out of the woods at the foot of the Mule Shoe, and quickly overran the outer defensive works. On the Confederate left, Walker's brigade, the "Stonewall Brigade," stood fast, and were nearly obliterated. Unfortunately, no one among the Federal commanders had expected the assault to be so successful, and there were no reserves standing by to exploit the opportunity, which might have cut Lee's army in half. General Gordon and others lead troops forward to plug the yawning gap in the line, and Lee tried to lead a counterattack, which resulted in another "Lee to the rear" incident. This was another bloody slaughter house on the long road to Richmond. Although Federal casualties were heavier, Lee lost nearly a quarter of his army there.
Lee anticipated Grant's next move, and fell back on the North Anna River. Grant had not yet been able to bring up his entire army, but he still outnumbered Lee considerably. However, the Federal troops were without bridging equipment or boats, and the attempts at an assault crossing were doomed. There was another bloodletting, but it was so paltry in comparison to the three weeks of battle which had preceded it that many histories of the war don't even discuss it in detail. Grant obligingly moved by his left again, to try to get around Lee's army, which lead him to the yard of that Justice of Peace, and yet another failure to trap "the Old Fox."
I think that Grant's frustration at not being able to win a conclusive battle is what lead him to order the ill-fated attack at Cold Harbor. I think he probably had an attitude of "just one more push" which would see the collapse of the Army of Northern Virginia. Those boys, however, were veterans, the ones who had held on for almost three years, and whose experience made them far more formidable than Grant was apparently giving them credit for. They dug themselves in, and without any guidance from engineers (none was needed) they laid out fields of converging fire from the heights above Cold Harbor, which doomed any assault. More than just "the Old Fox," the veteran troops of Lee's army defeated him.
In the entire "Overland Campaign," as it became known, Grant started with almost 120,000 troops (only about 100,000 were with the army at the beginning of the campaign), and suffered 55,000 casualties. He got more than 60,000 replacements, however. Lee started with 65,000 troops, and suffered about 30,000 casualties. But he only got 15,000 replacements, and about 8000 more troops from Beauregard's little army. Lincoln had indeed found the man who "understands the numbers."
(The exact figures for troops and casualties might be disputed, but not the overall effect.)