But only one Bragg, right?
Screw alternate history. Life follows baseball. 
Many Braggarts, one Bragg. 
The Union was handicapped by the fact that it inherited the command structure, minus defections to the rebels, in place at the time. Since America hadn’t been at war for twelve years, there hadn’t been much real ability to demonstrate war-time leadership qualities. And, as pointed out, the numbers commanded in the Mexican-American War were relatively small in comparison to the army commands involved in the Civil War; Gen. Winfield Scott had only 12,000 with him when he landed, and only about 8,000 effectives with him on his campaign inland.
Therefore, the Union command structure had a number of leaders who were simply there because they had done their time, not because they were good military commanders. An example was Brig. Gen. Irvin McDowell, commander of the Army of Northeastern Virginia at the First Battle of Bull Run. McDowell had no actual command experience at the time; the closest he had come to command was to be aide-de-camp to Gen. John Wool in Mexico. It’s not that McDowell was totally incompetent; the plans for First Manassas weren’t that bad, and if that battle had occurred later in the war, when the troops were properly trained and led, it might have succeeded. But he didn’t know how to command, and his subordinates didn’t, either. But McDowell was only in command because he had gotten friendly with Sen. Salmon Chase, who was now in the Cabinet, and who got him promoted to his command.
It took time during the war for the cream to rise to the top. I will point out that that was true in the South among the rebels as well. But the rebels did start with a greater number of capable generals (many of whom had served in command of units in the Mexican-American War), including P. G. T. Beauregard, McDowell’s opponent at Bull Run/Manassas. So they had an early advantage.
Someone else above has already pointed out that the Union might have done much better if the first real competent general it identified hadn’t turned out to be a complete egotist who was more interested in making himself the dandy of Washington D.C. than he was in actually fighting a war. General McClellan had shown a good grasp of tactics and a fighting spirit in the mountains.
I may be mistaken, but I seem to recall that a few of the better generals for the Union were Army mis-fits, who had at various times resigned to do other things. I’m pretty sure that applies to both Grant and Sherman, and I seem to remember that others fit that bill as well.
If I remember correctly, Lee’s Lieutenants has an appendix of all the Confederate high-ranking officers killed in the war. It was a long list. In many cases, an officer was killed in battle before he had a chance to really learn his job.
Even Grant and Sherman had to learn on the fly. For example they were both responsible for getting caught with their pants down at Shiloh. Though their talent shined through and they eventually partially redeemed themselves in the latter stages of the battle, that initial failure to properly watch for the enemy was very nearly disastrous.
For that matter if Albert Sidney Johnston hadn’t died in combat early in the battle it is possible they might still have lost. He seems to have been both much more aggressive and a bit more imaginative than his #2 Beauregard, who assumed full command after his death. Albert Johnston was another example of a highly respected senior commander who had defected to the south and was the highest ranked general to die in combat on either side. It is probable his early death hurt the Confederacy more than most, though hard to say if it would have made a serious difference.
I read somewhere that McClellan took to heart the deaths of his soldiers to an unhealthy (healthy?) level. Therefore, he wanted to make absolutely sure that before committing them to their deaths, he was in the absolute correct tactical and strategic position before acting.
The end result was that he always waited too long and got more of his men killed than necessary. He should have captured Richmond during the Peninsula Campaign in 1862. Indeed he was within sight of Richmond until Lee took over and shook his nerve. Lee should have been destroyed at Antietam. He was outnumbered 2-1 AND McClellan had found Special Order 191 which detailed Lee’s plans.
But McClellan hesitated and sent his men in piecemeal to be slaughtered.
I’ve always thought that McClellan’s failure wasn’t really his - or not all of it - but of the organization in which he was a part.
Simply put, he was a brilliant management General but not a great field General. If the union had the imagination to say early in the war, “George, your talents lie it getting a group of dumbasses trained, supplied and ready to fight. Do that here in Washington City. When you say they’re ready, you’ll hand them off to whichever field general - Meade, Grant, Whomever depending on the theatre - and we’ll get you a new batch to do your magic on.” things might have done differently.
Different people have different talents. But circumstance forced a square peg of McClelland into a round home of ‘He’s a general. He can do it all.’ that just didn’t work.
As for Grant and his superior numbers perhaps being responsible for his reputation as a field General? Grant was the one with the vision to finally say, “Fuck this. This has gone on long enough. I’m going to end thing thing if I have to bury the south in bodies to do it. I’m coming straight at Lee so he can’t respond. Cump, you march to Savannah and stop the south’s ability to prosecute the war.”
That’s not ‘rep by numbers’, that’s realistically seeing your advantages and using them.
I’ve often thought that’s one of the most human and realistic comments about army camaraderie you can find.
As summarised by Grant:
Which I always thought was a P. G. Wodehouse line, since he quoted it frequently. 
I agree with much of what Elvis, JB99 and Little Nemo have written.
Lee was so talented that he might have won the war for the U.S. before Lincoln had been able create the fragile political consensus that let him issue the Emancipation Proclamation - meaning slavery would persist, and perhaps lead to another war in a generation or two.
An early Grant biographer tried to put the best face on this, saying that yes, Grant was surprised at Shiloh, but he expected to be surprised.
Shiloh was the Civil War opposite of the Spanish Inquisition, then?
You have to admit that P.G.T. Beauregard was probably the coolest general officer name in the war on either side.
There was an American general named Jefferson Davis.
Totally belied by his results in the opening days of the war. The reason he was in Washington, D.C. was because he had obtained good results in the field in western Virginia. He was a perfectly fine field commander, as long as he wasn’t given time to second-guess himself.
Did he have independent command authority in those early situations, or was he subordinated to a superior?
It can make a psychological difference if you’re following orders, as opposed to acting on your own.
Winfield Scott was the commanding General of the United States Army in 1861. Old, fat, and unable to sit on a horse he was competent but in no way a field general. In any case, Lincoln replaced him after Bull Run. With McClellan.
Pretty much independent, IIRC. He was in command of the Department of Ohio, and I believe pretty much took direction only from Lt. Gen. Winfield Scott, the general-in-chief of the United States Army. His actions in Grafton, Phillipi and Rich Mountain were taken on his own initiative, to protect the vital Baltimore & Ohio RR.
At the time, he was one of the second highest-ranking officers in the Army (Maj. Gen.), behind only the general-in-chief.
What is your point? Lt. Gen. Winfield Scott was the general-in-chief of the United States Army. That person would never be expected to lead an army in battle. His plan for winning the war (derisively dubbed the “Anaconda Plan” by detractors) was almost exactly the strategy which eventually won the war: split the rebels by winning the Mississippi, while strangling them of supply along the coasts.
Of all the generals the Union had, Scott was probably one of the best. But he did have some idiosyncracies of his own, and he wasn’t dynamic enough to hold his position in the chain of decision-making. As a result, he got routinely bypassed by members of the Cabinet, by his junior officers, and by the President. His replacement by McClellan, or someone else with verve and elan, was probably inevitable.