Childhood Nightmare Fuel

Offhand, since unicorns are magical creatures that heal wounds / purify taint / nullify evil, they are very susceptible to becoming Mary Sue characters. "Oh, the kingdom is beset by poisonous vampire ooze monsters! Surely we shall all perish!" Then a unicorn shows up: everybody gets cured, the poisons are counteracted, and the ooze monsters dissolve like the Wicked Witch of the West. Kinda boring story-wise, innit? Of necessity, the unicorns either have to be flawed, usually aloof and uncaring; or the problem is greater than what they can counteract. That makes for good drama. Your example of The Last Unicorn is a case in point: almost all the unicorns are gone from the world, because none could withstand "the Red Bull." They met a greater power, and were vanquished, though not destroyed.

In the case of The Secret of NIMH, that was three former Disney animators (Bluth, Goldman, Pomeroy) escaping from the Mouse Works and launching their own feature animation meant to compete nose-to-nose with Disney. So, to make the audience sympathize with the mouse Brisby, Aurora (the studio's name before it was rebranded) added the dangerous mouse-killing cat Dragon, and the subversive villain Jenner. Three plot threads neatly intertwined, and each thread had a "scissors" element that threatened to unravel it all. Which is mighty fine story-crafting, lemme tell ya.

My top nightmare inducer, and I know I've said this before, is "Butterscotch And Soda," a very early Little Audrey cartoon. It was so scary that just reading about it makes me physically ill.