I don't believe that LotR's caused the same outcry from cetain groups as Harry Potter did. The major difference between the two is the use of magic. While "magic" exist in Middle-Earth it is not the same type of magic that is used in Harry Potter. The certain Christian groups that found Harry Potter so offensive claimed that it encouraged witchcraft and wizardry, both of which are forbidden.
So the analogy between Nazgul and Dementors while not in itself perfect, is not the cause of the outrage of Harry Potter. It is more based on the fact of the use of magic and magic wands and giving children the idea that it's okay to use magic. I think, myself, that they are a bit off but to each his own.
