The irony is that they can’t

Pretty sure I saw a bunch of these guys sucking people's souls out or whatever, during the battle, at least in the movie.

From the book, its ambiguous. Per Legolas to the hobbits, following the battle:

Faint cries I heard, and dim horns blowing, and a murmur as of countless far voices: it was like the echo of some forgotten battle in the Dark Years long ago. Pale swords were drawn; but I know not whether their blades would still bite, for the Dead needed no longer any weapon but fear

A lot of Tolkien's storytelling involves this kind of Word of Mouth recounting, such that it's hard to know whether you're getting real High Magic or just mythology passed down second hand.

However you slice it, I'd describe "literally scares you to death" as physical harm.