There is no real reason to think that. The Nazis used slave labor quite effectively.
Walt
There is no reason to think that it you are unfamiliar with the debate going on in the South at that time. While some vocal advocates of Slavery existed, most Southerners, unlike the Nazis, were deeply religous Christians, respected differing opinions and questioned the "particular institution". Besides, most Southerners did not own slaves and for them Slaves represented economic competition. And the industrial revolution was making slavery unattractive economically.
I do not accept your premise that the Nazis used slaves effectively. Slaves can find subtle ways to throw a monkey wrench into a complex process, and they did. Would you want to fight knowing your ammunition and weapons had been built by resentful forced labor. German artillery had a staggering percentage of duds, US almost never, according to Steven Ambrose of whomever he stole it from.
The Nazis did not even employ slaves unashamedly. The Nazis did all they could to hide the fact that they were using slave labor from the German people and the world.