I don't think the importance of this factor can be overestimated. It's pretty clear that no mortal men not in the possession of divine grace could resist the terror of the Nine its height. However, I think there's something rather interesting here that may have been missed.
To start off with, we need to go back to basics. Tolkien was a deeply devout Catholic. In Middle Earth's legendarium he wanted to create a mythic prehistory of the world which was still consistent with Christian and in particular Catholic teaching. He was vastly more subtle (and proportionately more effective) at doing this than his friend C.S. Lewis, but it's still pretty straightforward. Eru is the Christian God, not just figuratively, but literally, and the same goes for Morgoth as Lucifer. There is even an origin story of humanity in Middle Earth, the Tale of Adanel, which Tolkien only hinted at in the Silmarillion but is provided in full in one of the volumes of his collected writings. This is essentially the Eden story but with tribes of humans instead of Adam and Eve, and Morgoth as the Serpent, and Original Sin taking the form of humans being seduced by Morgoth's and eventually worshipping him in place of Eru, before some (the ancestors of the heroic Edain of the First Age who gave rise to the Númenóreans) repented.
We can also see the Christian theology underpinning Middle Earth reflected in the veneration that the Elves and more noble men give to Eru. They venerate him, but they have no organized religion because they predate even Abraham by eons, and positing an organized Abrahamic religion before Abraham would be deeply blasphemous! (In fact, the Noldor and Numenoreans predate the Flood, which may or may not in Middle Earth be the same thing as the Downfall of Númenor.) Indeed, Tolkien himself reckoned that the Fall of Barad-Dur was roughly six thousand years distant from modern days, putting the twentieth century somewhere between the Fifth to Seventh Age depending on how it is reckoned. As far as Tolkien would be concerned if you asked him, the War of the Ring was not a fantasy story to the Union soldiers, it is an event which happened in their ancient past.
Now it's worth asking the question, did anything significant occur in the intervening millennia between Sauron's downfall and the American Civil War?
To a devout Christian like Tolkien, the most important event in history happened in that time. It is an event which means every Union soldier, especially the faithful amongst them, carries with them a grace that even Elendil or the mightiest amongst the Elves did not possess. It is a grace which means that the terror of the Nazgûl, which is but merely the faded after-echo of that terrible fear and darkness which pursued the first Men who fled out of Hildórien, may frighten and alarm the Union soldiers, but cannot rob them of their wills. Their hearts have been vouchsafed by another, a king even mightier than Melkor or Manwë Súlimo.
They have been Washed in the Blood of the Lamb.
Well I have to say that this really is an interesting point to consider. Thank you for bringing a fresh viewpoint to this.