My thoughts:
Yes, this is PC B.S., but only by exaggeration, as there is some truth to the general idea. As was mentioned, they weren't a factor until late. Recall the comment that Shelby Foote made that the US was always going to win, but for much of the war they were fighting with one arm behind their back and all they had to do was bring that arm around. The colored soldiers were part of that arm, but not all of it. The north had more factory workers and farmers exempted than the south and they simply had the manpower.
One might be tempted to say, well bring those factory workers into the army and then replace them with freed blacks, but that would have blown the copperhead powderkeg apart. Whites would never want their jobs taken by blacks, because they might not get them back when the war was over. So in a lot of ways, the addition of colored manpower did help. If the North had never used them, it would have been hard, but not impossible. But one could say it was like the Atomic Bomb: it led to the end, and it made it less costly to some degree, but the conclusion was a foregone one by Gettysburg, if not sooner.
Dr. Watson's point about them hurting the south by refusing to work, etc. is true, but is somewhat overblown by some historians who want to credit blacks with an empowering act rather than being spectators, so they gravitate to that idea, as the black historian on the Civil War documentary did.
I think more important than what they denied to the south was what they provided to the North. I say this because many of the ancillary roles in the army that the North used them for, the south was loath to do so anyway. They used some, to be sure, but there simply wasn't as much make work in the southern army for them as there was in the far more logistically complex Union army.
I think overall, it's a sketchy argument to make with certainty. I will say that they were far more important in the frontier army than most people realize, and they were hugely important, although often overlooked, in the Spanish American war. It was "Black" Jack Pershing's colored troops who led the main assault on San Juan, and the Rough Riders only joined in the attack after they had taken the difficult, but smaller Kettle Hill.
I think that was the high point of colored soldiers, because by World War I, the army (ironically run by Pershing, who was very pro-colored troops, hence his nickname) was trying to move them away from most combat roles.
