I think the Nazgul failed for the same reason that the Eagles didn't fly the ring to Mount Doom.
We wouldn't have had as good a story then.
Seriously, I think readers (and the Fellowship?) need a taste of the power they face and the terror they will meet as a way of helping build suspense. We don't really know or understand what this terrible power is until as readers we 'experience' the Nazgul.*
*keeping in mind that Tolkien's version of Fairie is not as dark as some of the original fairy stories, tales and legends.